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GRADE IX
March 26
2019HIGHER ORDER THINKING QUESTIONS FROM
American Mathematics Competitions,
Canadian Mathematics Competitions,
Indian Examinations – JEE, RMO
MATHEMATICS
1
INDEX
No. Topic Page number
1 Real numbers 2 - 23
2 Factors 24 - 27
3 Polynomials 28 - 30
4 Coordinate Geometry 31 – 47
5 Linear Equations 48 - 49
6 Geometry 50 - 51
7 Axioms, Definitions, Theorems 52 - 81
8 Angles 82 - 107
9 Isosceles Triangles 108 - 133
10 Model proof 134 - 135
11 Parallelogram 136 - 149
12 Mid-segment Theorem 150 - 154
13 Area of parallelograms and triangles 155 - 186
14 Circle 187 - 210
15 Euclid’s 5th Postulate 211 - 215
16 Volume 216 - 224
Note: 1. American Competitions are AMC – 8, AMC – 10A, 10B, AMC – 12 A, 12 B.
Note: 2. Canadian Competitions are Gauss, Pascal, Cayley, Fermat, Euclid, and
Descartes.
Note 3. Indian examinations are JEE (Main, Advanced), Pre Rmo, RMO
In case of doubt in any problem, just Google the problem.
2
PROPERTIES OF REAL NUMBERS
1. The properties of operations: Here a , b and c stand for arbitrary numbers in a
given number system. The properties of operations apply to the rational number
system, the real number system, and the complex number system.
Associative property of addition
)()( cbacba
Commutative property of addition abba
Additive identity property of 0 aaa 00
Existence of additive inverses For every a there exists a so that
aaaa )(0)(
Associative property of multiplication )()( cbacba
Commutative property of
multiplication
abba
Multiplicative identity property of 1 aaa 11
Existence of multiplicative inverses For every 0a there exists a
1 so that
111
aaa
a
Distributive property of
multiplication over addition
cabacba )(
3
2. The properties of equality: Here a , b and c stand for arbitrary numbers in the
rational system, real or complex number systems.
Reflexive property of
equality
a = a
Symmetric property of
equality
If a = b , then b = a
Transitive property of
equality
If a = b and b =c , then a =c .
Addition property of equality If a = b , then cbca
Subtraction property of
equality
If a = b , then cbca
Multiplication property of
equality
If a = b , then cbca
Division property of equality If a = b , and 0c then
c
b
c
a
Substitution property of
equality
If a =b , then b may be substituted for a in any
expression containing a .
4
The properties of inequality: Here a , b and c stand for arbitrary numbers in the
rational or real number systems.
Exactly one of the following is true a < b or a = b or a >b .
If ba and cb then ca .
If ba , then ab .
If ba , then ba .
If ba , then cbca .
If ba , then cbca
If ba and 0 c then cbca .
If ba and 0 c then cbca .
If ba and 0 c thenc
b
c
a .
If ba and 0 c thenc
b
c
a .
Area of each colored triangle is 0.5 square unit. Hence the area of the inner square is 2
squared units. Since x is the side of the inner square, 22 x .
2
1
1
x
1
1
5
The side of the square and its diagonal are incommensurable line segments.They are
not both integral multiples of another segment.
EH = 2 - 1EG = 2 - 2
AC = 2 AB = AG = 1
12 - 1 3 -2 2
5 2 - 7
2 - 1 GC =
IM
N
P
O
J
L
K
E
F
H
G
DA
CB
6
Side of the square (s) Diagonal of the square (d = 2 s)
1 2
12 22
223 423
725 2710
21217 2427̀1
41229 24158
27099 140299
2392169 2239338
d = 2 s, always, however small
Reference: MAA Journal
7
For A series paper, the ratio of the longer side to the shorter side is 2
Two A5 size papers make one A4 size paper.
SURDS CAN BE EVALUATED TO ANY DEGREE OF ACCURACY USING
FRACTIONS
12
112
Therefore 12 =
212
1
Hence 122
112
………. (1) Let 12 = y. Substituting in (1)
yy
2
1 Replace y by
y2
1 in the RHS we get
b
a
a
b = 2
A7
A6
A5
A4
8
y
y
2
12
12
12
1
OR
...2
12
12
12
112
Hence we can approximate the value of 2 to any degree of accuracy by fractions. In
the above example 12
412 = 3.416…, correct to 2 places of decimals.
9
REAL NUMBERS
1. PASCAL – 2017(4): Which of the following is closest to 7?
A) 70 B) 60 C) 50 D) 40 E) 80
2. The rational number, which equals the number 357.2 with recurring decimal is
A) 1001
2355
B)
997
2379
C) 999
2355 D) none of these
3. Which of the following is not an irrational number?
A) 2 B) 3 C) 4 D) 5
4. The number whose decimal expansion is non terminating and non- recurring is
A) rational number B) irrational number
C) any real number D) none of these
5. What is the difference between the largest and smallest of the numbers in the list
0.023, 0.302, 0.203, 0.320, 0.032?
A)0.090 B) 0.270 C)0.343 D) 0.288 E)0.297
6. PASCAL – 2003(10): Which of the following numbers is the largest?
A) 3.2571 B) 2571.3 C) 5712.3
D) 7125.3 E) 1257.3
7. PASCAL – 2004(17): The value of 1.0 + 12.0 + 123.0 is
A) 343.0 B) 355.0 C) 53.0
D) 355446.0 E) 355445.0
8. When the numbers 607.5 , 760.5 , 5. 07, 5.076, 076.5 are arranged in increasing order,
the number in the middle is
10
A) 607.5 B) 760.5 C) 5. 07
D) 5.076 E) 076.5
9. CAYLEY-2011(8): If 8104.2 is doubled, then the result is equal to
A) 8204.2 B)
8208.4 C) 8108.4
D) 16104.2 E)
16108.4
10. AMC 12A – 2013(10): Let S be the set of positive integers n for which n
1 has the
repeating decimal representation ...,.0.0 abababab with a and b different digits.
What is the sum of elements of S?
A)11 B)44 C)110 D) 143 E)155
11. AMC Book - IV1974(20): Let
T= 25
1
56
1
67
1
78
1
83
1
;
then
A) T < 1 B) T = 1 C) 1 < T < 2
D) T=)25( )56( )67( )78( )83(
1
E) T > 2
12. Which list of numbers is written in increasing order?
A) 2011, 2011 , 22011
B) 2011, 22011 , 2011
C) 2011 , 2011, 22011
D) 2011 , 22011 , 2011
E) 22011 , 2011 , 2011
11
13. Book I -1958(5): The expression 22
1
22
122
equals
A) 2 B) 22 C) 22 D) 22 E) 2
2
14. Book I -1958(8): Which of these four numbers
2 , 3 8.0 , 4 00016.0 , 13 )09.0(1
is(are) rational:
A) none B) all C)the first and the fourth
D)only the fourth E)only the first
15. Book V-83(21): Find the smallest positive number from the numbers below
A) 11310 B) 10113 C) 13518
D) 261051 E) 512610
16. 10A – 2011(16): Which of the following is equal to 269269 ?
A) 3 2 B) 2 6 C) 2
27 D) 3 3 E) 6
Hint: Square the given expression
17. Contest Book V Page 9. Q 13: 532
62
equals
A) 532 B) 324 C) 5632
D) 2
352 E)
3
253
ANSWERS
1 C 2 C 3 C 4 B 5 E 6 D
7 D 8 E 9 B 10 D 11E 12 C
13 A 14D 15 D 16 B 17 A
12
18. How to Solve it, G.POLYA Page Number 234(4): Among Grandfather’s papers a
bill was found:
The first and last digit of the number that obviously represented the total price of those
fowls are replaced here by blanks, for they have faded and are now illegible.
What are the two faded digits and what was the price of one turkey?
Answer: 3, 2 and the price of one turkey is 5.11 Hint: The 5 digit number is divisible by both 8, 9.
19. Represent as a common fraction the decimal_____28137.24 . Answer:
99999
2428113
20. Express _
74.0 in the form ofq
p, where p and q are integers. Answer:
90
43
21. Write one rational number and two irrational numbers between 7
5and
11
9.
22. Represent 3.5 on the number line.
72 turkeys $ _67.9_
15.3
5.3
13
23. Represent 13 on the number line.
24. The decimal expansion of 7
1 is the repeating decimal 142857.0 . What digit occurs in
the 2014th
place after the decimal point? Answer: 8
2
313
13
10
2
14
RATIONAL AND IRRATIONAL – I
From common core, USA
1. Write three rational numbers.
____________________________________________________
2. Explain what a rational number is in your own words.
_____________________________________________________
3. Write three irrational numbers.
____________________________________________________
4. Explain what an irrational number is in your own words.
____________________________________________________
5. In the figure the rectangle has sides lengths a and b.
Decide if it is possible to find a and b to make the statements below true.
If you think it is possible, give values for a and b.
If you think it is impossible, explain why no values of a and b will work.
a) The perimeter and area are both rational numbers.
____________________________________________________
b) The perimeter is a rational number, and the area is an irrational
a
b
15
number.
____________________________________________________
c) The perimeter and area are both irrational numbers.
____________________________________________________
d) The perimeter is an irrational number, and the area is a rational
number.
____________________________________________________
6. Decide whether you think each statement is always, sometimes or
never true.
Always True: Explain why
Sometimes true: Write an example for which it is true and an example for
which it is false.
Never true: Explain
a) The product of two rational numbers is irrational.
___________________________________________
b) The product of a rational number and an irrational number is
irrational.
_____________________________________________
c) The sum of two irrational numbers is irrational.
_______________________________________
16
d) The diagonal of a square is irrational.
___________________________________
e) The sum of two rational numbers is rational.
________________________________________
f) The sum of a rational number and an irrational number is
irrational.
_____________________________________________________
g) The circumference of a circle is irrational.
________________________________________
h) The product of two irrational numbers is irrational.
____________________________________________
Answers:
2) A rational number is a number that can be written as a fraction of integers.
4) Cannot be written as a fraction of integers.
5) b) 64 , 63 c) 5 , 7 d) 5 , 3 5
6 b) Sometimes. 0 5 = 0, 3 5
c) Sometimes. 523 , 52
d) Sometimes. a = 2, 5a
f) Irrational numbers are non-terminating, non-repeating. Adding repeating part to non-
repeating decimal will not make it repeating. Adding a rational number to an irrational
causes the irrational number to shift on the number line.
g)
4r
h) 2 , 8 . 3 , 8 .
17
RATIONAL AND IRRATIONAL - II
1. For each of the numbers below, decide whether it is rational or irrational.
Explain your reasoning.
a) 27
3 b) 273 c)
27
3 d)
327 327
e) 1 f) 0.45 g) 6.0 h)
3
2. Change the fractions into decimals.
a) 3
2 b)
2
3 c)
11
3 d)
6
5
3. Complete the following explanation to convert 38.0 into a fraction.
Let
38.0x
100 x = ………………….
10x =………………….
100 x – 10 x = ………………….
So x = ………………….
4. Change the repeating decimal 45.0 into a fraction.
5. a) Change 1.5 and 94.1 into a fraction.
b) Change 1.2 and 91.1 into a fraction.
Write your observation.
________________________________________
18
6. Decide whether you think each statement is always, sometimes or never
true.
Always True: Explain why
Sometimes true: Write an example for which it is true and an example for
which it is false.
Never true: Explain
a) If you divide one irrational number by another, the result is
always irrational.
b) If you divide a rational number by an irrational number, the
result is always irrational.
c) If the radius of a circle is irrational, the area must be irrational.
d) Between two irrational numbers there is an irrational number.
e) If the circumference of a circle is rational, then the area is
rational.
f) If the area of a circle is rational, then the circumference is
rational.
g) Between two rational numbers there is an irrational number.
h) An expression containing both 6 and is irrational.
i) The hypotenuse of a right triangle is irrational.
19
ANSWERS for question 6 a) to 6 i)
a) False. 13
3
b) False. If the rational number is zero, the quotient is always rational. Otherwise the result is always irrational.
c) False. If
7r ,then area = 49
d) True. Find the difference d between the two rationals. Add 2
dto the smaller number.
Find the smaller rational number. Add the digits of . e) Never true. f) Never true.
g) True. Find the difference d between the two rationals. Add 2
dto the smaller number.
Find the smaller rational number. Add the digits of .
h) Sometimes true. 6 + is irrational. But
24
6is rational.
i) Sometimes true
20
SCIENTIFIC NOTATION
A number written in the scientific notation has the form
na 10
where 10 1 a and n is an integer.
1. PROPERTIES OF INTEGER EXPONENTS
If m and n are integers and a and b are real numbers we have the following:
i) Product Rule nmnm aaa
ii) Power of a Power Rule mnn
m aa
iii) Power of a Product Rule mmm baab
iv) Power of a Quotient Rule m
mm
b
a
b
a
, 0b
v) Quotient Rule nm
n
m
aa
a
vi) Zero Exponent Rule 10 a , 0a
vii) Negative Exponent Rule m
m
aa
1 , 0a
2. RATIONAL EXPONENTS
i) If m and n are natural numbers b any real numbers we have the following:
nm
mn
nm
b
b
b1
1
)(
)(
21
Note: b cannot be negative when n is even.
ii) If m and n are natural numbers b any real numbers we have the following:
n
m
n
m
b
b1
Note: b cannot be negative when n is even.
1. Express in the simplified form.
a) 32 b) 4 9616 yx
c) 3
4
27 d) 3
2
8
e) 2
3
9
Answers:
a) 24 b) 4 2 2 yxxy c) 81 d) 4 e)
27
1
2. Simplify:
a) 20455 b) 12275327
c) 542183324245
d) 84 2222 e) 40240
Answers: a) 517 b) 311 c) 2764
d) 8 128 2 e) 10215 4
22
3.
4. Express as roots of rational numbers:
a) 80204 b) 2233 2435
c) 3
4
8
1
2
1
4
3
9 4 3 2
d) 14 37 142 15 3 5
Answers: a) 160 b) 62 29 c) 6 3 18 d)
14 3 53
5. Rationalize the denominator of:
a) 35
1
b)
35
35
c) 2453
5223
d)
321
1
e) 532
2
Answers:
a) 2
35 b) 154 c)
13
10 6
d) 4
622 e)
6
1563
23
6. Which is greater 2 1532 or 2 2310
7. G. Polya Mathematical Discovery Book II - P25: Which is greater 113 or
85 ?
Hint: Go on squaring both expressions. The second number is greater.
10 + 3 2 215 + 2 3 2
3 2
3 23 2
10
1010
10
18
3 2
10
6 5
6 56 5
6 5
15
15
15 15
2 3
15
122 3
2 3
2 3
24
FACTORS
(a + b)2 = a2 + 2ab + b2
ab
ab
b2
a2
b
b
b
b
aa
a
a
= a2 + b2 + c2 + 2ab + 2bc + 2ca
c
c
c
c
b
ba
a
ac
bc
ab
c2
b2
a2
(a + b + c)2
ac
bc
ab
25
1. 1952(3): The expression 33 aa equals:
A)
2
2 11
1
aa
aa B)
2
2 11
1
aa
aa
C)
2
2 12
1
aa
aa D)
2
21
11a
aa
a
E) none of these
2. 1955(13): The fraction 22
44
ba
ba is equal to:
A) 66 ba B) 22 ba C) 22 ba
D) 22 ba E) 22 ba
3. 1955(3): Of the following expressions the one equal to 33
11
ba
ba is:
A) 22
22
ab
ba
B)
33
22
ab
ba
C)
33 ab
ab
= (x + 2) (2x + 1)
2+x
+2x 1
2 x2 + 4 x + x + 2
1
1
22
xx
2x
2x
26
D) ab
ba 33 E)
ba
ba
33
4. 12B-11 (15): How many positive two-digit integers are factors of 1224
(A) 4 (B)8 (C) 10 (D) 12 (E) 14
5. 10A - 1995(14): If 5)( 24 xbxaxxf and f(–3)=2, then f(3)=
A) –5 B) –2 C) 1 D) 3 E) 8
6. Contest Book VI, Page 20, Q 20: The sum of all real x such that
333
6422442
xxxx
is
A) 23
B) 2 C) 25
D) 3 E) 27
7. Contest Book I, 1953(18): One of the factors of 44 x is:
A) 22 x B) 1x C) 222 xx
D) 42 x E) none of these
ANSWERS
1 A 2 C 3 B 4 D 5 E 6 E 7 C
8. Pre RMO-2013(11) Three real numbers x, y, z are such that, 1762 yx ,
142 zy and 222 xz . What is the value of 222 zyx ?
27
Answer: 14, Add the 3 expressions and complete it as a perfect square.
9. Pre RMO-2015(1) Let ABCD be a convex quadrilateral with AB = a, BC =b, CD = c
and DA =d. Suppose
dacdbcabdcba 2222
and the area of ABCD is 60 square units. If the length of one of the diagonals is 30
units, determine the length of the other diagonal.
Answer: 4, Complete the expression as sum of 4 perfect squares. ABCD is a rhombus.
10. RMO -2014(2): Find all real numbers x and y such that
)12(2
12 22 yxyx Hint: Make perfect SQUARES in x and y
Answer: x = 1, y = 0.5
11. RMO– 2014(1): Three positive real numbers a, b, c are such that
04445 222 bcabcba . Can a, b, c be the lengths of the sides of a triangle?
Justify your answer. Hint: Make perfect SQUARES in a, b and c
Answer: No, a = 4c > b + c
12. Simplify and evaluate: (American Mathematical Journal)
36333633
2013120141201420131201412014 Answer: 4
Hint: Let y 2014 , ay 3 1 , by 3 1
28
POLYNOMIALS
1. If 12 93 23 xkxx is divisible by 3x ,then it is also divisible by:
A) 43 2 xx B) 43 2 x C) 43 2 x
D) 43 x E) 43 x
2. When
5x , x
x1
and 3
321
xx
are multiplied, the product is a polynomial of degree
A) 2 B) 3 C) 6 D) 7 E) 8
ANSWERS
1 C 2 C
3. Find the remainder when 1223 234 xxxx is divided by 2x . Answer: 67
4. Find the remainder when 79442 71788178636993700 xxxxx is divided by
2x . Answer: 666
5. Find the remainder when xxxxx 9254981 is divided by xx 3 .
Answer: 5 Hint: Let the remainder be CBxAx 2
.x = 0, 1 and –1 are zeros.
6. If ax is a factor of 22 32 baxx , then prove that ba .
29
7. Find a, b and c if cbxaxxx 234 3 is divisible by )1)(1)(2( xxx .
Answer: a = 1, b = –3, c = –2
8. Factorize completely: 12872 234 xxxx .Answer: (x + 1) (x + 2) (x – 2) (x – 3)
9. If 12 x is a factor of 16)1(4 234 xaxxax then find the value of a.
Answer: a = 13
10. If the polynomial 7212173846 xxxx is divided by another polynomial
143 2 xx , the remainder comes out to be )( bax , find a and b.
Answer: a = 1, b = 2
11. If 322 xx is a factor of 9 24 xax find the value of ‘a’. Answer: a = –10
12. Show that (x + 1) and (x – 3) are the factors of the polynomial 9122 23 xxx .
13. a) Determine the value of ‘m’ such that (x – 5) is a factor of the polynomial
50 163)( 23 xmxxxP Answer: m = –5
b) Determine the value of ‘a’ such that (x – 4) is a factor of the polynomial
2827 2)( 23 xxaxxP Answer: m = –13
14. Find the values of ‘a’ and ‘b’ so that the polynomial 623 bxaxx is completely
divisible by 342 xx . Answer: a = 1, b = 4
15. If ax is a factor of 10522 2 xaxx , find a. Answer: a = 2
30
16. Find all the zeros of the polynomial 120423434 xxxx , if two of its zeros are
2 and –2. Answer: 2, –2, 5 and –6
17. For what value of k, (–4) is a zero of the polynomial ?)22(2 kxx Answer: k = 9
31
COORDINATE GEOMETRY
1. In the diagram, ABC is isosceles and its area is 240. The y-coordinate of A is
A) 6 B) 12 C)18
D) 24 E) 48
2. Triangle ABC has vertices A (1, 2), B (4, 0) and C (1, – 4). The area of ABC is
X
Y
B (0, 0) C (20, 0)
A
Y
XO(0, 0)
A(1, 2)
C(1, -4)
B( 4, 0)
32
A) 18 B) 12 C) 8
D)10 E) 9
3. What is the area of rectangle ABCD?
A) 15 B) 16 C) 18
D)30 E) 9
Y
X
C(4, 2)
A
O
D(4,5)
B(-1, 2)
33
4. In the diagram, what is the area of ABC ?
A) 36 B) 54 C) 108
D)72 E) 48
5. In the diagram, ABC is right angled at B, AB is horizontal and BC is vertical. What
are the coordinates of B?
X
Y
A(4, 9)
B (0, 0) C (12, 0)
34
A) (5, 2) B) (5, 0) C) (5, 1)
D)(4, 1) E) (1, 5)
6. Points with coordinates (1, 1), (5, 1) and (1, 7) are three vertices of a rectangle. What
are the coordinates of the fourth vertex of the rectangle?
A) (1, 5) B) (5, 5) C) (5, 7)
D) (7, 1) E) (7, 5)
X
Y
A(2, 1) B
C(5, 5)
X
Y
(1, 1)
B (0, 0)
(1, 7)
(5, 1)
35
7. The point in the xy- plane with co-ordinates (1000, 2012) is reflected across the line
y = 2000. What are the co-ordinates of the reflected point?
A) (998,2012) B) (1000, 1988) C) (1000, 2024)
D) (1000, 4012) E) (1012, 2012)
8. 10B – 2008(14): Triangle OAB has O= (0, 0), B = (5, 0) and A in the first quadrant. In
addition, 90ABO and 30AOB . Suppose that OA is rotated 90
counterclockwise about O. What are the coordinates of the image of A?
A)
5 ,
3
310 B)
5 ,
3
35 C) 5 ,3
D)
5 ,
3
35 E)
5 ,
3
310
Hint: In a 30o – 60
o – 90
o triangle, the side opposite to 30
o angle is always half of the hypotenuse.
9. 12A - 2004(13): A set S of points in the xy-plane is symmetric about the origin, both
the co-ordinate axes, and the line y = x. If (2, 3) is in S, what is the smallest number of
points in S?
5
2y
y
y
x30°
A
B (5, 0)O
36
A) 1 B) 2 C) 4 D) 8 E) 16
10. 10A- 2011(9): A rectangular region is bounded by the graphs of the equations ay ,
by , cx and dx , where a, b, c and d are all positive numbers. Which of the
following represents the area of this region?
A) bdbcadac B) bdbcadac C) bdbcadac
D) bdbcadac E) bdbcadac
11. AJHSME -1991(10): The area in square units of the region enclosed by
parallelogram ABCD is
A) 6 B) 8 C)12 D) 15 E) 18
12. 10A – 2016(16): A triangle with vertices A (0, 2), B (–3, 2) and C (–3, 0) is reflected
about the x-axis, then the image CBA is rotated counterclockwise about the origin
by 90 to produce CBA . Which of the following transformations will return
CBA to ABC ?
A) Counterclockwise rotation about the origin by 90 .
(0, 0)
y
x(3, 0)
(4, 2)D
BA
C
37
B) Clockwise rotation about the origin by 90 .
C) Reflection about the x-axis.
D) Reflection about the line xy .
E) Reflection about the y-axis.
13. 12B - 2004(9): The point (–3, 2) is rotated 90 clockwise around the origin to the
point B. Point B is then reflected in the line y = x to point C. What are the coordinates
of C?
A) (–3, –2 ) B) (–2, –3) C) (2, –3)
D) (2, 3) E) (3, 2)
14. 8 -2015(18): A triangle with vertices as A = (1, 3), B = (5, 1) and C = (4, 4) is plotted
on a 5 6 grid. What fraction of the grid is covered by the triangle?
y
x
(-3, 2)
C(-3, 0)
(0, 2)
C'' B''
A''
A'B'
B A
O
38
A) 6
1 B)
5
1 C)
4
1 D)
3
1 E)
2
1
15. In the diagram, rectangle ABCD has area 70 and k is positive. The value of k is
A) 8 B) 9 C) 10
D)11 E) 12
y
x
C
A
B
Y
X
C(k, -3)
A(1, 4)
O
D(k, 4)
B(1,-3)
39
16. 12A - 2005(25): Let S be the set of all points with coordinates (x, y, z), where x, y and
z are each chosen from the set {0, 1, 2}. How many equilateral triangles have all their
vertices in S?
A) 72 B) 76 C) 80 D) 84 E) 88
17. JEE(Main) – 2011(78): The number of points, having both co-ordinates as integers,
that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0), is
A) 901 B) 861 C) 820 D)780
18. The number of integral points (integral point means both the coordinates should be
integer) exactly in the interior of the triangle with vertices (0, 0), (0, 21) and (21, 0) is
(A) 133 (B) 190 (C) 233 (D) 105
z
y
x
2
2
2
(0, 0, 1) (0, 1, 1)
(1, 1, 1)
(1, 0 , 0) (1, 1, 0)
(0, 1, 0)
(1, 0, 1)
40
19. 8-2001(11): Points A, B, C and D have these coordinates: A (3, 2), B (3, –2) C (–3, –
2) and D (–3, 0). The area of quadrilateral ABCD is
A) 12 B) 15 C) 18 D) 21 E)24
(2, 17)
(1, 19)
(0, 21)
(0, 0)(21, 0)
(1, 1) (1, 19)
(2, 2) (17, 2)
y
xD (- 3, 0)
C (-3, - 2) B (3, - 2)
A (3, 2)
41
20. 8 -1996(17): Figure OPQR is a square. Point O is the origin, and the point Q has
coordinates (2, 2). What are the coordinates for T so that the area of triangle PQT
equals the area of square OPQR?
A) (–6 , 0) B) (–4, 0) C) (–2, 0)
D) (2, 0) E) (4, 0)
21. 10B – 2009(17): Five unit squares are arranged in the coordinate plane as shown, with
the lower left corner at the origin. The slanted line, extending from (a, 0) to (3, 3),
divides the entire region into two regions of equal area. What is a?
(2, 2)
Y
XT P
R Q
O
42
A) 2
1 B)
5
3 C)
3
2
D) 4
3 E)
5
4
22. PASCAL-2015(21): Each diagram shows a triangle, labeled with its area.
Y
X(a, 0)
(3, 3)
y
x(0, 0)
Area= m
(1, 4)
(4, 1)
(0, 0)
y
x
(4, 4)
(3, 0)
(0, 1)Area= n
43
What is the correct ordering of the areas of these triangles?
A) m < n < p B) p < n < m C) n< m < p
D) n < p < m E) p< m < n
ANSWERS
1 D 2 E 3 A 4 B 5 C 6 C 7 B 8 B
9 D 10 A 11 B 12 D 13 E 14 A 15 D 16 C
17 D 18 B 19 C 20 C 21 C 22 D
23. EUCLID – 2001(1b): In the diagram, points P (3, 2), Q (11, 2) and R (11, k) form a
triangle with area 24, where k > 0. What is the value of k?
(0, 0)
y
x
(4, 3)
(2, 0)
(0, 4)
Area= p
44
Answer: k = 8
24. EUCLID – 2009(1c): The point (k, k) lies on the line segment AB shown in the
diagram. Determine the value of k. Answer: k = 2
O
y
x
R(11, k)
Q(11, 2)P(3, 2)
B(8, -4)
A(0, 4)
y
x
O
45
25. Find the area of the shaded region.
Answer: 41
25. CAYLEY - 2006(20): The line 94
3 xy crosses the x – axis at P and the y-axis at
Q. Point srT , lies on the line segment PQ. The area of triangle POQ is 3 times
the area of triangle TOP. Determine the values of r and s, the coordinates of T.
Y
X(22, 0)(6, 0)
(0, 0)
(0, 15)
(0, 11)(2, 10)
46
Answer: r = 8, s = 3
s
r
y
x
y = 3
4x + 9
Q
PO
T(r, s)
47
26. Three of the vertices of square ABCD are located at A (0, 3), B (4, 0) and C (7, 4).
Determine the area of square ABCD. Area: 25
27. The side of a square is 24 unit long and its diagonals are parallel to the coordinate
axes. If the center of the square is (3, 5), find the coordinates of its vertices.
Answer: (3, 9), (7, 5), (3, 1)), (-1, 5).
A(0, 3)
(7, 4)
(4, 0)x
y
B
C
y
xO
(3, 5)
4 2
EB
A
C
D
48
LINEAR EQUATIONS
1. 12A - 2004 (3): For how many ordered pairs of positive integers (x, y) is 1002 yx ?
A) 33 B) 49 C) 50 D) 99 E) 100
2. The number of solution-pairs in positive integers of the equation 50153 yx is:
A) 33 B) 34 C) 35 D) 100 E) none of these
3. 10A - 1995(10) : The area of the triangle bounded by the lines xy , xy and
6y is
A) 12 B)12 2 C) 24 D) 24 2 E) 36
4. 1971(3): If the point (x, –4)lies on the straight line joining the points (0, 8) and
(–4, 0)in the xy-plane, then x is equal to
A) –2 B) 2 C) –8 D) 6 E) –6
5. Given the line 1553 yx and a point on this line equidistant from the coordinate
axes. Such a point exists in
A) none of the quadrants B) quadrant I only C) quadrants I, II only
D) quadrants I, II, III only E) Each of the quadrants
Hint: Draw the graphs of y = x, y = –x.
6. 12B -2009 (9): Triangle ABC has vertices A = (3, 0), B = (0, 3) and C, where C is on
the line 7 yx . What is the area of ABC ?
A) 6 B) 8 C) 10 D) 12 E) 14
49
ANSWER
1 B 2 A 3 E 4 E 5 C 6 A
7. For how many positive integers (x, y) is 1003 yx ? Answer: 33
8. There is one pair (x, y) of positive integers for which 1935 yx . What are the
values of x and y? Answer: (2, 3)
9. A pen costs Rs.11 and a notebook costs Rs.13. Find the number of ways in which a
person can spend exactly Rs.1000 to buy pens and notebooks. Answer: 10001311 yx
10. If P(x, –4) lies on the line joining A (0, 12) and B (3, 0), find x. Answer: x = 4
y
xB (3, 0)
A (0, 12)
50
GEOMETRY
Acknowledgement: Unknown
JUST FOR FUN
In Egypt, land of the Pharaohs, was its birth,
Where it was used to measure the earth,
You see, every year the Nile River would flood,
And wash away absolute tons of mud
And then it was no longer too very much clear,
To distinguish whose land was whose my dear.
They invented the surveyor who found their boundary
But did nothing about cleaning the laundry.
The building of pyramids about 3000 B.C.
Used several geometric principles, see?
Then from Greece came Thales a man,
Who was the first geometrician,
He and others learned from Egyptian priests,
Where geometry still has not ceased.
After Thales came the great Plato,
Who made the idea of proof grow and grow.
Aristotle, a Greek who really rates.
Noted the difference between axioms and postulates.
51
Then came the all important EUCLID.
Probably never heard of him. You did?
Euclid studied under the one and only Plato.
Remember, the one who made theorems grow and grow.
He was the most successful text book writer,
You must understand compared with us, he was much brighter.
He wrote thirteen books called THE ELEMENTS
which I know, don’t tell me, is in the wrong tense.
THE ELEMENTS are the basis for geometry teaching,
And it seems to me to be the mostly on deductive reasoning
This poem tells the ancient history of geometry through Euclid
I hope you enjoyed it ‘cause I sure did.
52
Acknowledgement: Geometry, Scott, Foresman
Undefined terms: Point
Line
Plane
STEPS IN DEDUCTIVE PROOF
Undefined terms
Axioms
Definitions
Theorems
53
LINES AND ANGLES
Axiom: A line contains at least two points. A plane contains at least three
non-collinear points. Space contains at least four non-coplanar, non-
collinear points.
Axiom: Two points are contained in one and only one line. (Two points
determine a line)
Theorem: If two lines intersect, then they intersect in one and only one
point.
Axiom: Three points are contained in one and only one plane. (Three
noncollinear points determine a line)
1. Definition: Collinear points are points that are contained in one line.
Non collinear points
B
A
C
54
2. Definition: Coplanar points (or lines) that are contained in one plane.
Theorem: Two intersecting lines lie in one and only one plane.
3. The word between is used to describe collinear points only.
Definition: A segment RT is the set of points containing R, T and all the
points between R and T.
.
Ruler Postulate: For every pair of points, there is a unique positive real
number called the distance between the two points.
Segment addition axiom: If point S is between points R and T, then
RTSTRS
4. Definition: A ray RT is the set of points containing RT and all the points S
such that T lies between R and S.
A
B
C
P
TR
55
Axiom: On any ray, there is exactly one point at a given distance from the
endpoint of the ray.
5. Definition: BA and BC are opposite rays if and only if A, B and C are
collinear and B is between A and C.
6. Definition: A midpoint of a segment R is a point S between R and T such
that STRS .
Axiom: A segment has exactly one midpoint.
TR
CA B
SR T
56
7. Definition: A bisector of a segment is a set of points whose intersection
with the segment is the midpoint of the segment.
8. Definition: An angle is the union of two non-collinear rays which have the
same end point.
(The rays are called the sides, common end point is the vertex.)
Protractor Postulate: For every angle there is a unique real number r,
called its degree measure, such that 1800 r .
SR T
57
9. Definition: The interior of angle APB is the intersection of two planes: the
side of PA containing B and the side of PB containing A.
10. Definition: The exterior of an angle is the set of points in the plane which
do not belong to the interior of the angle or to the angle itself.
P A
B
Exterior
P
A
B
58
11. Definition: A bisector of an
APC is a ray PB such that B is in the interior
of APC and BPCmAPBm .
Axiom: If B is the interior of angle APC, then APCmBPCmAPBm .
Axiom: An angle has exactly one bisector.
12. Definition: A right angle is an angle whose measure is 90.
Theorem: All right angles are equal.
Definition: An acute angle is an angle whose measure is less than 90.
P
A
C
B
ABC = 90°
B
C
A
59
Definition: An obtuse angle is an angle whose measure is greater than 90,
but less than 180.
13. Definition: Complementary angles are two angles whose measures have a
sum of 90. Each angle is called a complement of the other.
Theorem: Complements of equal angles are equal.
P
A
C
P A
C
B
DC
B
D
A
60
Theorem: Complements of same angles are equal.
14. Definition: Supplementary angles are two angles whose measures have a
sum of 180. Each angle is called a supplement of the other.
Theorem: Supplements of equal angles are equal.
Theorem: Supplements of same angles are equal.
15. Definition: Adjacent angles are two coplanar angles with a common side
and no common interior points.
P
C
B
P A
B
61
Definition: A linear pair of angles is a pair of adjacent angles whose non
common sides are opposite rays.
Axiom: The angles in a linear pair are supplementary.
Theorem: If one angle of a linear pair is a right angle, then the other angle
is also a right angle.
Theorem: If the angles in a linear pair are equal, then the lines containing
their sides are perpendicular.
16. Definition: Vertical angles are two angles whose sides form two pairs of
opposite rays.
P A
C
B
P AC
B
62
Theorem: Vertical angles are equal.
17. Definition: Perpendicular lines are lines that intersect to form a right
angle.
Theorem: Two perpendicular lines intersect to form four right angles.
l
m
l m
63
18. Definition: A perpendicular bisector of a segment is a line which is
perpendicular to the segment and contains its midpoint.
19. Definition: Parallel lines are lines that are coplanar and do not intersect.
Segments and rays are parallel if the lines containing them are parallel.
20. Definition: A transversal is a line that intersects two or more coplanar
lines in distinct points.
SR T
l
m
l m
64
Corresponding angles 21
Alternate interior angles. 32
Axiom: Two coplanar lines cut by a transversal are parallel if and only if a
pair of corresponding angles are equal.
Theorem: If two parallel lines are cut by a transversal, then alternate
interior angles are equal.
(lines Alt, s =)
Theorem: If two lines are cut by a transversal so that a pair of alternate
interior angles are equal, then the lines are parallel. (Alt, s = lines .)
l
m
n
3
2
1
65
Theorem: If two parallel lines are cut by a transversal, then interior angles
on the same side of the transversal are supplementary. (lines int s
on the same side are supp )
Theorem: If two lines are cut by a transversal so that a pair of interior
angles on the same side of the transversal are supplementary, then the
lines are parallel. (int s on the same side are supp lines )
Theorem: If a transversal is perpendicular to one of the two parallel lines,
then it is perpendicular to the other.
l
m
n2
1
1 = 2
l
m
n
1 + 2 = 180°
2
1
66
Theorem: In a plane, if two lines are perpendicular to the same line, then
they are parallel.
Properties of parallel lines.
Corresponding angles are equal.
Alternate interior angles are equal.
Interior angles on the same side of the transversal are supplementary.
A line perpendicular to one line is perpendicular to another.
To prove lines are parallel, show that:
The lines are coplanar and do not intersect.
Two corresponding angles are equal.
Two alternate interior angles are equal.
Two interior angles on the same side of the transversal are supplementary.
A transversal is perpendicular to each of two given lines.
21. The following theorems require indirect proof:
Theorem: If two lines intersect, then they intersect in one and only one
point.
Theorem: Through a point not on a given line, there is exactly one line
parallel to the given line.
Theorem: In a plane, if two lines are parallel to the same line, then they
are parallel to each other.
Theorem: In a plane, through a point on a given line, there is exactly one
line perpendicular to the given line.
Theorem: In a plane, a segment has exactly one perpendicular bisector.
67
Theorem: Through a point not on a given line, there is exactly one line
perpendicular to the given line.
Theorem: If two planes are perpendicular to the same line, then the
planes are parallel.
22. Definition: Skew lines are lines that are not coplanar.
23. Definition: A line and a plane that intersect are perpendicular if and only if
the given line is perpendicular to every line in the given plane that passes
through the point of intersection.
24. Definition: Two planes, or a line and a plane, are parallel if and only if they
do not intersect.
P
68
TRIANGLES
1. Definition: A triangle is the union of three segments determined by three
noncollinear points.
2. Definition: An acute triangle is a triangle with three acute angles.
Definition: A right triangle is a triangle with a right angle.
B C
A
Acute
B C
A
RightLeg
Leg
Hypotenuse
B
C
A
69
Definition: An obtuse triangle is a triangle with an obtuse angle.
Definition: An equiangular triangle is a triangle with three equal angles.
CBA
Obtuse
B C
A
A
B C
70
3. Definition: A scalene triangle is a triangle with no two sides equal.
Definition: An isosceles triangle is one with at least two equal sides.
Base angles B and C
B C
A
Vertex
LegLeg
Base
B C
A
71
Definition: An equilateral triangle is a triangle with three equal sides.
AB = BC = CA
Theorem: The sum of the measures of the angles of a triangle is 180.
Theorem: If two angles of one triangle are equal to two angles of another
triangle, then the remaining angles are equal.
Theorem: The acute angles of a right triangle are complementary.
Theorem: Each angle of an equiangular triangle has measure 60.
Theorem: The measure of an exterior angle of a triangle is equal to the
sum of the measures of its remote interior angles.
A
B C
3 2
1
3 2
1
72
Theorem: The measure of an exterior angle of a triangle is greater than
the measure of either of its remote interior angles.
4. Definition: A polygon is the union of three or more coplanar segments
such that each segment intersects exactly two other segments, one at
each endpoint, and no two intersecting segments are collinear.
Convex polygon,
Consecutive angles A, B. Consecutive sides AB, BC
1 + 2 = 3
3
2
1
B C
A
D
E
F
AB
C
D
73
Non-convex polygon.
5. Definition: An equiangular polygon is a polygon whose angles are equal.
Note: Measure of each angle is 120O
E
F
A
B
C
D
2
6
4
5
3
1
E
F
C
BA
D
74
6. Definition: An equilateral polygon is a polygon whose sides are equal.
7. Definition: A regular polygon is a polygon that is both equiangular and
equilateral.
Theorem: The sum of the measures of the angles of a convex polygon of n
sides is 180 )2( n .
C
B D
A
CB
A D
F E
75
Theorem: The sum of the measures of the exterior angles of a convex
polygon, one angle at each vertex, is 360.
8. Definition: Two triangles are congruent if and only if there is a
correspondence between the vertices such that each pair of
corresponding sides and corresponding angles are equal.
Theorem: Congruence of triangles is reflexive, symmetric and transitive.
Included angle for two sides of a triangle is the angle whose sides contain
the two sides of the triangle.
Included side for two angles of a triangle is a side whose endpoints are the
5.3 cm 5.7 cm
6.4 cm
5.3 cm 5.7 cm
6.4 cm
D
B C
A
E F
B C
A
76
vertices of the angle.
Axiom: If three sides of one triangle are equal to the corresponding parts
of another triangle, then the triangles are congruent. SSS
Axiom: If two sides and the included angle of one triangle are equal to the
corresponding parts of another triangle, then the triangles are congruent.
SAS
D
E F
4.5 cm 6.1 cm
5.9 cm
4.5 cm6.1 cm
5.9 cm
D
B C
A
E F
77
Axiom: If two angles and the included side of one triangle are equal to the
corresponding parts of another triangle, then the triangles are congruent.
ASA
Theorem: If two angles and the side opposite one of the angles in one
triangle are equal to the corresponding parts of another triangle, then the
triangles are congruent. AAS
Theorem: If the hypotenuse and a leg of one right triangle are equal to the
corresponding parts of another triangle, then the triangles are congruent.
RHS Note: This is side – side – angle rule applicable only to right triangles.
1. Definition: An angle bisector is the segment from any vertex to the
opposite side on the ray bisecting the angle.
4.4 cm
5.5 cm
4.4 cm
5.5 cm
F
A
B
CDE
5.0 cm5.0 cm
F
A
B
CD E
78
Definition: A median is the segment from any vertex to the midpoint of
the opposite side.
Definition: An altitude is the segment from any vertex to perpendicular to
the line containing the opposite side.
2. Definition: The distance from a point to a line (or plane) not containing the
point is the length of the perpendicular segment from the point to the line
D
B C
A
DB C
A
D BC
A
79
(or plane).
Theorem: If two sides of a triangle are equal, then the angles opposite
those sides are equal.
Theorem: If two angles of a triangle are equal, then the sides opposite
those angles are equal.
Theorem: An equiangular triangle is equilateral.
Theorem: An equilateral triangle is equiangular.
TRIANGLE INEQUALITY
3. Theorem: If the lengths of two sides of a triangle are unequal, then the
measures of the angles opposite those sides are unequal in the same
order.
l
B
A
80
Theorem: If the measures of two angles of a triangle are unequal, then
the lengths of the sides opposite those angles are unequal in the same
order.
Theorem: If two angles of a triangle are unequal, then the greater angle
has longer side opposite to it.
Theorem: The sum of the lengths of any two sides of a triangle is greater
than the length of the third side.
AC > AB 1 > 2
21
B C
A
AC > AB1 > 2
21
B C
A
81
Hinge Theorem: If two sides of one triangle are equal to two sides of
another triangle and the included angles are unequal, then the lengths of
the third sides are unequal in the same order.
4. Theorem: If two sides of one triangle are equal to two sides of another
triangle and the lengths of the third sides are unequal, then the measures
of the angles included between the equal sides are unequal in the same
order.
5. Theorem: If ACBCAB , then the points A, B and C are collinear and B is
between A and C.
Theorem: The perpendicular segment from a point to a line is the shortest
segment from the point to the line.
BA + AC > BC
B C
A
82
ANGLES
1. In the diagram PQR is 90 and RQS is 50 greater than PQS . Then
PQS , in degrees, is
A) 70 B) 50 C) 45 D) 40 E) 20
2. The angle through which the hour hand of a clock travels between 9 : 20 a.m. and
11 : 50 a.m. on the same morning is
A) 45 B) 60 C) 75
D) 85 E) 80
3. What is the number of degrees in the smaller angle between the hour hand and the
minute hand on a clock that reads seven o’clock?
A) 50 B) 120 C) 135 D) 150 E) 165
4. At 2: 15 o’clock, the hour and minute hands of a clock form an angle of:
A) 30 B) 5 C) 5.22 D) 2
17 E) 28
QR
SP
83
5. 8 -2003(20): What is the measure of the acute angle formed by the hands of clock at
4:20 a.m?
A) 0 B) 5 C) 8 D) 10 E) 12
6. 8 -2012(3): On February 13 The Oshkosh Northwester listed the length of daylight as
10 hours and 24 minutes, the sunrise as 6:57 A.M, and the sunset as 8: 15 P.M. The
length of daylight and the sunrise were correct, but the sunset was wrong. When did
the sun really set?
A)5 :10 PM B)5 :21 PM C)5 :41 PM
D)5 :57 PM E)6 :03 PM
7. In diagram, the value of x is
A) 25 B) 30 C) 50 D)55 E)20
80°x°
55°50°B A
C
84
8. In diagram, C lies on the line segment BD. What is the value of x?
A) 50 B) 55 C) 75 D)100 E)105
9. In diagram, AD is parallel to BC. What is the size of ABC ?
A) 116 B) 168 C) 138 D) 144 E) 122
125°
75°
x°
B C
A
D
128°
2x°x°
B C
A D
85
10. What is the measure of the largest angle in ABC?
A) 144 B) 96 C) 120 D) 60 E) 108
11. CAYLEY - 2006(8): A rectangle is drawn inside ABC, as shown. If 22BEF
and 65CDG , then the size of BAC is
A) 87 B) 90 C) 93 D) 104 E) 82
3x°
6x° x°
A
BC
F G
D
A
B C
E
86
12. In the diagram, ABC is a straight line. The value of x is
A)27 B) 33 C) 24 D) 87 E) 81
13. If AD is a straight line segment and E is a point on AD, determine the measure of
DEC .
A) 20 B) 12 C) 42 D) 30 E) 45
57°
2x°21°21°
x°
BA C
(10x - 2)°(3x + 6)°20°
EDA
C
B
87
14. In the diagram, the value of x is
A)130 B) 120 C) 110 D) 100 E) 80
15. In the diagram, B, C and D lie on a straight line, with 100ACD , xADB ,
xABD 2 and yBACDAC . The value of x is
A)10 B) 45 C) 30 D) 50 E) 20
x°
60°
40°
C
D F
A
E
B
100°
y° y°
x°2x°
CB D
A
88
16. In the diagram, the value of x is
A)40 B) 60 C)100 D)120 E)80
17. In the diagram, AB is a straight line. The value of x is
A)67 B) 59 C) 62 D)40 E)86
40°
120°
x°
y°y° y°
59° x° 140°BA
89
18. In the diagram the value of x is
A) 100 B) 65 C) 80 D)70 E)50
19. In the diagram, BCD is a straight line. The value of x is
A) 72 B) 44 C) 58 D) 64 E) 52
20. In diagram, what is the value of x?
A) 65 B) 75 C) 85 D) 95 E) 105
130°
x°
136° 64°
x°
B
A
C D
x°
45°50°
90
21. In triangle ABC, 72B . What is the sum, in degrees, of the other two angles?
A) 144 B)72 C) 108 D) 110 E) 288
22. CAYLEY – 2012(14): In the diagram, ABC and DEF overlap so that CEBF
forms a straight line segment. What is the value of x?
A)10 B)20 C)30 D) 40 E)50
23. CAYLEY - 1997(16): A beam of light from point B, reflects off a reflector at point F,
and reaches point G so that GF is perpendicular to EA. Then x is
72°
A
B C
x°30°
50°
C FE B
D
A
91
A) 32 B) 37 C) 45 D) 26 E) 38
24. Which of the following statements is not true?
A) A quadrilateral has 4 sides.
B) The sum of the angles in a triangle is 180 .
C) A rectangle has four 90 angles.
D) A triangle can have two 90 angles.
E) A rectangle is a quadrilateral.
x°
x°
26°
CE AD
F
B
HG
92
25. In the diagram, the value of x is:
A) 72 B)158 C) 108 D) 138 E) 162
26. In a triangle ABC, A is 21 more than B , and C is 36 more than B . The
size of B is
A) 20 B) 41 C) 62 D) 46 E) 56
27. In a ABC, BA 3 and CB 2 . The measure of B is
A) 10 B) 20 C) 30 D) 40 E) 60
28. In the diagram, BDC is a straight line. What is the measure of DAC
A) 27 B) 47 C) 48
D) 65 E) 67
48°
60°
x°
D
E B
C
F
A
67°
38°48°
B C
A
D
93
29. In the diagram, line segments AB and CD intersect at E. The value of x is
A) 30 B) 20 C) 40 D) 50 E) 35
30. In the diagram E lies on the line segment DC. The value of x is
A) 120 B) 130 C) 135 D)140 E)150
31. FERMAT – 2003(14): In the diagram, ABC, CDE, AEF and BDF are all straight
lines. The value of ba is
x°
110°
50°
EA
B
C
D
50°
x°
E
A
B
D
94
A) 70 B) 55 C) 80 D) 90 E) 75
32. In the diagram, ABC, ADE and BDE are straight lines. The value of x is
A) 75 B)85 C) 95 D) 125 E) 155
33. In the diagram ABC is a straight line. What is the value of x?
125°
b°
a°
55°
D
A C
F
B
E
x°
125°30°A B
D
C
F
E
(x+14)° 110°
3x°
A
BC D
95
A) 19 B)62 C) 21.5 D) 24 E) 32
34. In the diagram, three line segments intersect as shown. The value of x is
A)40 B)60 C)80 D) 100 E)120
35. In the diagram, the value of x is
A)20 B)60 C)70 D) 40 E)50
36. In triangle ABC, the value of x + y is
A) 104 B) 76 C) 180 D) 90 E)166
60°
x°40°
x°
50°70°
104° y°x°B C
A
96
37. In the diagram, AB is parallel to CD, F and G lie on CD, and H and I are the points of
intersection of AB with EF and EG, respectively. If 120FHI and 112IGD ,
what is the measure of HEI ?
A) 52 B) 56 C) 60 D) 64 D) 68
38. In the diagram, the value of a + b in degrees, is
A) 70 B) 200 C) 130 D) 100 E)160
39. In the diagram, point E is on side AC of triangle ABC and BCD is a straight line
segment. The value of x is
IH
C D
A
E
F G
B
ba 130°
70°
97
A)55 B)70 C)75 D)60 E) 50
40. If two straight lines intersect as shown, then yx is
A) 0 B) 40 C) 80 D)60 E) 100
41. 12A-09(10): In quadrilateral ABCD, AB = 5, BC = 17, CD = 5, DA = 9, and BD is an
integer. Find BD. Topic: Triangle Inequality
30°
x°x°(180 - x)°
E
DB
A
C
y°x°
40°
98
A) 11 B)12 C) 13 D) 14 E) 15
42. 12B- 11(10): Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on the side
AB so that CMDAMD . What is the degree measure of AMD ?
A) 15 B) 30 C) 45 D) 60 E) 75
43. 12B- 2008(8): Points B and C lie on AD. The length of AB is 4 times the length of
BD, and the length of AC is 9 times the length of CD. The length of BC is what
fraction of the length of AD?
A) 36
1 B)
13
1 C)
10
1 D)
36
5 E)
5
1
44. 10 B – 2011(7): The sum of two angles of a triangle is 5
6of a right angle, and one of
these two angles is 30 larger than the other. What is the degree measure of the largest
angle in the triangle?
A) 69 B) 72 C) 90 D) 102 E)108
45. 10A - 2000(5): Points M and N are the midpoints of sides PA and PB of PAB. As P
moves along a line that is parallel to side AB, how many of the four quantities listed
below change?
a) The length of the segment MN.
b) The perimeter of PAB.
c) The area of PAB
d) The area of trapezoid ABNM.
C B
D
A
99
A)0 B) 1 C) 2 D) 3 E) 4
46. 10 A - 1999(7): What is the largest number of acute angles that a convex hexagon can
have?
A) 2 B) 3 C) 4 D) 5 E)6
47. 12A-2002(4): Find the degree measure of an angle whose complement is 25% of its
supplement.
A) 48 B) 60 C) 75 D) 120 E) 150
48. 12B-2007(7): All sides of the convex pentagon ABCDE are of equal length, and
90BA . What is the degree measure of E ?
A) 90 B) 10 8 C) 120 D) 144 E) 150
49. AMC 8- 2000 (13): In triangle ABC, we have ACBBAC and 36ABC . If CE
bisects ACB , then AEC =
NM
A B
P
100
A) 36 B) 54 C) 72 D) 90 E) 108
50. AJHSME 1997(12): 18021
43 . Find 4
A) 20 B) 25 C) 30 D) 35 E) 40
51. AJHSME-1996(24): The measure of angle ABC is 50 , AD bisects angle BAC, and
DC bisects angle BCA. The measure of angle ADC is
E
A C
B
40°
70°
4
321
101
A) 90 B) 100 C) 115 D) 5.122 E) 125
52. AMC 8 – 1999(21): The degree measure of angle A is
A)20 B) 30 C) 35 D)40 E) 45
53. If A is four times B , and the complement of B is four times the complement of
A , then B =
A) 10 B) 12 C) 15 D) 18 E) 5.22
50°
A C
D
B
100°
110°
40°
102
54. In the diagram, AD is parallel to EF and points B and C lie on AD. Also xABE ,
)50(xEBC and )25(xEFC .
What is the measure of FCD ?
A) 115 B) 140 C) 135
D) 130 E) 120
55. In the diagram, D is the midpoint of AB, 30CDB and 15BAC . The
measure of ABC is
A) 75 B) 65 C) 60 D) 80 E) 85
56. FERMAT – 2016(20): In the diagram, ABCD represents a rectangular piece of paper.
The paper is folded along a line EF so that 125EFD . When the folded paper is
flattened, points C, D have moved to points C’ and D’, respectively, and EC’ crosses
AF at G. The measure of AGE is
(x + 25)°
(x - 50)°x°
B C
E F
DA
BD
C
A
103
A) 110 B) 100 C) 95 D) 105 E) 115
57. CAYLEY-2013(19): In the diagram, PQ is parallel to RS. Also, Z is on PQ and X is
on RS so that 20YXS and 50ZYX , what is the measure of QZY ?
A) 30 B) 20 C) 40 D) 50 E) 60
58. In the diagram, points Q and R lie on PS and 38QWR . If
xTQWTQP , yVRWVRS , and U is the point of intersection
of TQ extended and VR extended, then the measure of QUR is
D'
C'
125°
G F D
CB
A
E
R S
P
Y
QZ
X
104
A) 71 B) 45 C) 76 D) 81 E) 60
ANSWER
1 E 2 C 3 D 4 C 5 D 6 B 7 A 8 A
9 A 10 E 11 A 12 A 13 C 14 D 15 E 16 E
17 A 18 C 19 A 20 C 21 C 22 B 23 A 24 D
25 E 26 B 27 D 28 A 29 A 30 D 31A 32 C
33 D 34 C 35 B 36 D 37 A 38 A 39 E 40 E
41 C 42 E 43 C 44 B 45 B 46 B 47 B 48 E
49 C 50 D 51 C 52 B 53 D 54 B 55 A 56 A
57 A 58 A
38°
y°y°x°
x°
U
P S
W
Q R
TV
105
59. In figure AB is parallel to CD. Find y.
Answer: 55o
60. AB is parallel to DC. Find x, y. Answer: x = 37o, y = 53
o
59. In the diagram, points B and C lie on AD. What is the value of x? Answer : x = 110o
60. In the diagram, point B is on AC, point F is on DB, and point G is on EB. What is the
value of utsrqp ?
y°
x°x°
70°C D
A B
65°
28°
y
x
D
A B
C
x°
60°
130°A D
E
B C
106
Answer: 360o
61. AIME (2) - 2006(1): In a convex hexagon ABCDEF, all six sides are congruent, A
and D are right angles, and B , C , E and F are congruent. The area of the
hexagonal region is 122116 . Find AB. Answer: 46
62. In figure, AB is parallel to CD and EF is parallel to GH. Find x.
Answer: x = 75o
u°
t°
s°r°
q°
p°A C
D E
B
F
G
x°
105°
C D
B
FH
GE
A
107
63. AB is parallel to CD. Find x, y.
Answer: x = 50o, y = 75
o
64. What is the measure of the obtuse angle formed by the hands of clock at 9:20?
Answer: 160o
y
x 125°
50°
C D
BA F
E
108
ISOSCELES TRIANGLES
1. ABCD is a square and EDC is an equilateral triangle. The number of degrees in angle
AEB is Hint: EC = BC
A) 90 B) 120 C) 135
D) 150 E) none of these
2. CAYLEY - 2007(8): In diagram, triangles ABC and CBD are isosceles. The
perimeter of CBD is 19, the perimeter of ABC is 20 and length of BD is 7. What
is the length of AB?
E
A
D
B
C
109
A) 5 B) 6 C) 7 D)8 E)9
3. In the diagram, AB = AC and 65ACB . The value of x is
A) 45 B) 30 C) 50 D)60 E)40
4. In the diagram, ACD, BCE are straight lines and AB = AC, CD = CE.
What is the value of x?
y°
y°
x°
x°
C
A
B
D
x°65°
D
B C
A
110
A) 45 B) 50 D) 55 E) 60 E) 65
5. In the diagram, DE is parallel to FG, AC = BC and 50DAB . The value of x is
A) 40 B) 65 C) 25 D) 100 E) 80
x°
40°B
C
E
A
D
x°50°
BG
CF
DE
A
111
6. In the diagram, line 1L is parallel to line 2L and BA = BC. The value of x is
A)35 B) 30 C) 37.5 D) 45 E) 40
7. CAYLEY - 2002(9): In the diagram, ABCD and DEFG are squares with equal side
lengths, and 70DCE . The value of y is
L2
L1x°
70°
CB
A
112
A) 120 B) 160 C) 130 D) 110 E) 140
8. In the diagram, A is on CD so that BA bisects CBD . Also AC = AB, yBDC 2
and yBAC 3 . The measure of BAC is
A) 90 B) 108 C) 120 D) 60 E) 72
y°
A
BF
G
EC
D
3y°
2y°
x°
x°
A
C B
D
113
9. CAYLEY - 2007(15): In the diagram, if ABC and PQR are equilateral, then
CXY equals
A) 30 B) 35 C) 40 D) 45 E) 50
75°65°
X
Y
Q
A
C
B P
R
114
10. In the diagram, ABC is a straight line segment and BD = BE. Find x.
A)28 B) 38 C)26 D)152 E)45
11. In the diagram, points C and D lie on BE. Also triangle ABE is isosceles with AB =
AE. Triangle ACD is isosceles with AC = AD. Find x.
A)11 B) 28 C)17 D)31 E)34
3x°
x°
76°
D E
A
B
C
34°
62°
x°
EC DB
A
115
12. In the diagram, a point E lies on the line segment AB, and triangles, AED and BEC
are isosceles. Also DEC is twice ADE . What is the size of EBC ?
A) 75 B) 80 C) 60 D) 55 E) 45
13. In the diagram, ABC is an equilateral triangle. If AB =4x and AC = x + 12, what is
the value of x?
A)48 B)16 C) 4 D) 32 E)12
14. A regular pentagon ABCDE has all its sides and angles equal. If the shaded pentagon
is enclosed by squares and triangles as shown, what is the measure of angle GEF ?
70°
C
D
BA E
x + 124x
CB
A
116
A) 72 B) 158 C) 108 D) 138 E) 162
15. In the diagram, DA = CB. What is the measure of DAC ?
A) 70 B) 100 C) 95
D) 125 E) 110
16. In right triangle ABC, AX = AD and CY = CD, as shown. The measure of
XDY is
xF
G
E
D
C
BA
70°40°
55° B
DA
C
117
A) 35 B) 40 C) 45
D) 50 E) not determined by this information.
17. In the diagram, CB = CD. The measure of CAD is
A) 45 B) 75 C) 40
D) 50 E) 25
18. CAYLEY - 2015(14): In the diagram, ABC is isosceles with AC = BC and ACD
is isosceles with AD = CD = x. Also, the perimeter of ABC is 22, the perimeter of
ACD is 22, the perimeter of ABCD is 24. What is the value of x?
Y
D
B C
A
X
85°35°
70°
C D
AB
118
A) 7.5 B) 6.5 C) 7 D) 6 E) 8
19. In the diagram, AB = AC and BD = CD. What is the measure of ABD ?
A) 45 B) 30 c) 60 D) 75 E) 15
20. In the diagram, C lies on AE, 40CED and CD = DE and ABC is equilateral.
The value of x is
x
C B
A
D
30°
60°
B C
D
A
119
A) 50 B) 60 C) 80 D) 90 E) 100
21. In the diagram, QUR and SUR are equilateral triangles. Also, QUP , PUT
and TUS are isosceles triangles with PU = QU = SU = TU and
QP = PT= TS. The measure of UST , in degrees, is
A) 50 B) 54 C) 60 D) 70 E) 80
22. In the diagram, ABCD is a square and ABE is an equilateral triangle. What is the
measure of CAE ?
40°x°
B
A EC
D
P
S
U
Q
R
T
120
A) 90 B) 105 C) 120 D) 150 E) 75
23. In triangle ABC, the value of x + y is
A) 104 B) 76 C) 180 D) 90 E)166
E
C
BA
D
y°x° 104°B
DC
A
121
24. In the diagram, if 48BED , what is the measure of BAC ?
A) 60 B) 42 C) 48 D) 66 D) 84
25. In the diagram, triangle ABE is equilateral and BCDE is a square. The measure of
angle BCA, in degrees, is
A) 10 B) 15 C) 20 D) 25 E) 30
A
E
B
C
D
A
D
EB
C
122
26. In the parallelogram, the value of x, is
A) 30 B) 50 C) 70 D) 80 E) 150
27. 8 – 2009(19): Two angles of an isosceles triangle measure 70 and x . What is the
sum of the three possible values of x?
A) 95 B) 125 C) 140 D)165 E)180
28. Triangle ABC is an isosceles triangle with AB = BC. Point D is the midpoint of both
BC and AE and CE is 11 units long. Triangle ABD is congruent to triangle ECD.
What is the length of BD?
A) 4 B) 4.5 C) 5 D) 5.5 E) 6
x°
150°
80°
E
D
CA
B
123
29. 10A-02(23): Let A, B, C and D lie on a line, in that order, with AB = CD and BC =
12. Point E is not on the line, and BE = CE = 10. The perimeter of AED is twice
the perimeter of BEC . Find AB.
A) 2
15 B) 8 C)
2
17 D) 9 E)
2
19
30. 12 A 2010(8): Triangle ABC has ACAB 2 . Let D and E be on AB and BC,
respectively, such that ACDBAE . Let F be the intersection of segments AE
and CD, and suppose that CFE is equilateral. What is ACB ?
A) 60 B) 75 C) 90 D) 105 E) 120
Hint: Find angles FAC
31. 12 A 2007(C and ADC are isosceles with AB = BC and AD = DC. Point D is inside
triangle ABC, 40ABC , and 140ADC . What is the degree measure of
BAD ?
12
10 10
DCBA
E
x
x
E
F
D
A
B C
124
A) 20 B) 30 C) 40 D) 50 E) 60
32. AMC 8-2005(15): How many different isosceles triangles have integer side lengths
and perimeter 23?
A) 2 B) 4 C) 6 D) 9 E) 11
33. In figure, BO bisects CBA , CO bisects ACB , and MN is parallel to BC. If AB =
12, BC = 24, and AC = 18, then perimeter of AMN is
A) 30 B) 33 C) 36 D) 39 E) 42
34. 8 -2000(24) : If 20A and AGFAFG , then DB
140°
40°
A C
B
D
NM O
B C
A
125
A) 48 B) 60 C) 72 D) 80 E) 90
35. In the diagram, ABC is isosceles with AB = AC. Also, point D on AC so that AD =
BD = BC. The measure of ABD is
A) 18 B) 30 C) 45
D) 36 E) 60
F
G
B
C
D
A
E
D
B C
A
126
36. In triangle ABC, AB =AC. If there is a point P strictly between A and B such that AP
= PC = CB, then A =
A) 30 B) 36 C) 48 D) 60 E) 72
ANSWERS
1 D 2 D 3 E 4 D 5 E 6 A 7 E 8 B
9 C 10 B 11 A 12 A 13 C 14 A 15 B 16 C
17 A 18 D 19 E 20 C 21 A 22 B 23 D 24 D
25 B 26 C 27 D 28 D 29 D 30 C 31 D 32 C
33 A 34 D 35 D 36 B
37. Determine ‘a’ for which the triangle is isosceles.
Answer: a = 5, 4
B
P
A
C
5a - 16
3a - 83a - 6
127
38. In the diagram, point P is inside quadrilateral ABCD. Also DA = DP = DC and AP =
AB. If xCDPADP 2 , )5(xBAP , and )510( xBPC , what is
the value of x? Answer x = 13o
39. In the diagram, ABCD is a square, ABE is equilateral, and AEF is equilateral.
What is the measure of DAF ? Answer x = 30o
40. In the diagram, the following information is known about ABC . The point D is on
side AC and the point E on side AB. Also, CD = EA, AD =BE. 80ABC ,
30ADE . Find all the remaining angles.
P
D C
AB
F
E
C
D
A
B
128
Answer: 80C , 20A , 130AED
41. ACEF is a rectangle with FE =4, FA = 7, AB = 1, CD = 4. Find .CBDABF
Answer: 135CBDABF , use Pythagoras theorem
80°
30°
E
BC
A
D
3
1
4
3
4
7
EF
A CB
D
129
42. RMO – 2001(5): In a triangle ABC, D is a point on BC such that AD is the internal
bisector of A . Suppose CB 2 and CD = AB. Prove that 72A .
Hint: Prove DCEBE A . Prove triangle AED is isosceles and consider exterior angles of the
equal sides.
F
E
DB C
A
130
THE PENTAGON PROBLEM
(from Common Core Lessons, USA)
This pentagon has three equal sides at the top and two equal sides at the bottom.
Three of the angles have a measure of 130 .
Figure out the measure of the angles marked x and explain your reasoning.
43. Method I: Draw a line down the middle of the pentagon.
x°
130°
130°
x°
130°
130°
x°
130°
130°
x°
131
44. Method II
Use the exterior angles of the pentagon to figure out the measure of x.
45. Method III:
Draw a line that divided the pentagon into a trapezoid and a triangle. Angle x has also
been cut into two parts labeled a and b.
x°
130°
130°
x°
130°
bb
aa
130°
130°
130°
132
46. Method IV: Divide the pentagon into three triangles.
Hint: 1) Use the fact that sum of 4 angles of a quadrilateral is 360 o. 2) Sum of the exterior angles of a
polygon is always = 360 o. 3) Interior angles on the same side of a transversal =180
o and property
of isosceles triangle. 4) Sum of angles of 3 triangles is 540 o
The diagram is made up of four regular pentagons that are all the same
size.
47. Find the measure of angle AEJ.
Show your calculations and explain your reasons.
x°
130°
130°
x°
130°
K
L
E D
C
B
J
M
N
FG
AH
I
133
___________________________________________________________
___________________________________________________________
___________________________________________________________
48. Find the measure of angle EJF.
Explain your reasons and show how you figured it out.
___________________________________________________________
___________________________________________________________
___________________________________________________________
49. Find the measure of angle KJM.
Explain how you figured it out.
___________________________________________________________
___________________________________________________________
134
Acknowledgement: Internet
EUCLID’S ELEMENTS
PROPOSITION 16: If any side of a triangle is extended, then the
exterior angle of a triangle is always greater than either of the
opposite interior angles.
Produce BC to D Postulate 2
Let E be the midpoint of AC By proposition 10.
AE = CE Property of midpoint
Draw BE Postulate 1
Extend BE to G Postulate 2
4
3
21
F
E
B C
A
D
G
135
With center E draw a circle of radius EB intersecting BG at F Postulate 3
BE = EF Property of midpoint
Draw CF Postulate 1
21 Vertically opposite angles are equal Proposition 15
AEB CEF by proposition 4
34 C.P.C.T
DCA > 3 by axiom 5
DCA > 4 Axiom 5
Axiom 5 (Common notion) the whole is greater than the part.
Proposition 10. Any line segment can be bisected into two congruent segments.
136
PARALLELOGRAM
1. Definition: A parallelogram is a quadrilateral in which both pairs of
opposite sides are parallel.
Theorem: Both pairs of opposite sides of a parallelogram are equal.
Theorem: Both pairs of opposite angles of a parallelogram are equal.
Theorem: The diagonals of a parallelogram bisect each other.
Properties of Parallelograms
Opposite sides are parallel.
Opposite sides are equal.
Opposite angles are equal.
Diagonals bisect each other.
Theorem: The distance between two given parallel lines is constant.
Theorem: If two sides of a quadrilateral are parallel and equal, then the
quadrilateral is a parallelogram.
CD
BA
C
A B
D
137
Theorem: If both sides of a quadrilateral are equal, then the quadrilateral
is a parallelogram.
Theorem: If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram.
Ways to prove a quadrilateral is a Parallelogram.
Show that:
Both the pairs of opposite sides are equal.
One pair of opposite sides are parallel and equal.
Both pairs of opposite sides are equal.
The diagonals bisect each other.
2. Definition: A rectangle is a parallelogram with four right angles.
C
A B
D
O
C
A B
D
C
A B
D
138
Theorem: A parallelogram is a rectangle if and only if its diagonals are
equal.
3. Definition: A rhombus is a parallelogram with four equal sides.
Theorem: A parallelogram is a rhombus if and only if its diagonals are
perpendicular.
C
A B
D
D
A C
B
139
Theorem: A parallelogram is a rhombus if and only if each diagonal
bisects a pair of opposite angles of the parallelogram.
B
CA
D
B
CA
D
140
4. Definition: A square is rectangle with four equal sides.
Theorem: If a segment joins the midpoints of two sides of a triangle, then
it is parallel to the third side, and its length is one half the length of the
third side.
Theorem: If three or more parallel lines cut off equal intercepts on one
DC
AB
D E
B C
A
141
transversal, then they cut off equal intercepts on every transversal.
5. Definition: A trapezoid is a quadrilateral with exactly one pair of opposite
sides parallel.
Parallel sides are called bases and non parallel sides are called legs.
6. Definition: The median of a trapezoid is the segment joining the midpoints
of the legs of the trapezoid.
FG = GH
AB = BC
BG
H
F A
C
Leg
Base
Leg
Base
D
A B
C
142
Theorem: The median of a trapezoid is parallel to the bases and its length
is one half the sum of the lengths of the bases.
7. An isosceles trapezoid is a trapezoid whose legs are equal.
Theorem: Each pair of base angles of an isosceles trapezoid are equal.
Theorem: The diagonals of an isosceles trapezoid are equal.
8. Definition: If exactly one diagonal of a quadrilateral is a perpendicular
bisector of the other, the figure is called a kite.
HG
CD
A B
AD = BC
B
D C
A
EA C
D
B
143
9. Definition: A locus is the set of points, and only those points, that satisfy a
given condition.
Theorem: In a plane, the locus of points equidistant from two given points
is the perpendicular bisector of the segment joining the two points.
Theorem: In a plane, the locus of points equidistant from the sides of an
angle is the bisecting ray of the angle, excluding its endpoint.
AP = BP
A B
P
AP = CP
A
C
B
P
144
CLASSIFYING QUADRILATERALS
1. Complete the blank spaces below with the word ‘ALL’, ‘Some’ or ‘No’ to make the
statements about quadrilaterals correct, giving reasons for your word choice. Your
reasons can include diagrams.
a) _________ rectangles are squares.
Reason for your choice of word.
__________________________________________________________
This is done for you.
Some rectangles are squares. A square has all the properties of a rectangle with the
additional property of four congruent sides.
b) _________ rhombuses are parallelograms.
Reason for your choice of word.
__________________________________________________________
c) _________ rectangles are parallelograms.
Reason for your choice of word.
__________________________________________________________
d) _________ parallelograms are squares.
Reason for your choice of word.
__________________________________________________________
e) _________ squares are rhombuses.
Reason for your choice of word.
__________________________________________________________
2. Which of the following quadrilaterals must have at least one pair of congruent sides?
Circle all that apply.
RECTANGLE SQUARE TRAPEZOID PARALLELOGRAM RHOMBUS
145
Explain your answer.
__________________________________________________________
3. Which of the following quadrilaterals diagonals must bisect each other at right
angles? Circle all that apply.
RECTANGLE SQUARE TRAPEZOID PARALLELOGRAM RHOMBUS
Explain your answer.
__________________________________________________________
146
PARALLELOGRAMS
1. 12B – 2007(11): The angles of a quadrilateral ABCD satisfy DCBA 432 .
What is the degree measure of A , rounded to the nearest whole number?
A) 125 B) 144 C) 153 (D 173 E)180
2. 12A- 2013(9):In triangle ABC, AB =AC = 28 and BC =20.Points D, E and F are on
sides AB, BC and AC respectively, such that DE and EF are parallel to AC and AB
respectively. What is the perimeter of parallelogram ADEF?
A) 48 B) 52 C) 56 D) 60 E) 72
ANSWERS
1 D 2 C
F
D
B C
A
E
147
3. ABCD is a parallelogram. From A and B perpendiculars AP, BQ are drawn to meet
CD or CD produced. Prove that AP = BQ.
4. E and F are the midpoints of AB and AC, two sides of the ABC . P is any point on
BC. AP cuts EF at Q. Prove that AQ = PQ.
5. E and F are the midpoints of the sides AB and CD respectively of the parallelogram
ABCD. Prove that AECF is a parallelogram.
P Q
B
D C
A
Q
FE
B C
A
P
F
E B
D C
A
148
6. ABCD is a parallelogram and its diagonals intersect at O. Through O a straight line is
drawn intersecting AB in P and CD in Q. Prove that OP = OQ.
7. ABCD is a parallelogram. The bisectors of the angles A and C meet the diagonal BD
in P and Q respectively. Prove that APB CQD .
8. In the quadrilateral ABCD, AB = CD; also ABC = BCD . Prove that AD and BC
are parallel.
Q
O
B
D C
A P
PQ
B
D C
A
C
A D
B
149
9. In figure ABCD is rhombus. H is on BC and K is on CD such that AB = AH = HK =
KA. Determine the measure of BAD .
10. RMO – 2005(1): Let ABCD be a convex quadrilateral; P, Q, R, S be the midpoints of
AB, BC, CD, DA respectively such that triangles AQR and CSP are equilateral. Prove
that ABCD is a rhombus. Determine its angles.
Hint: Prove the shaded triangles are equilateral, isosceles. Find the measure of angles A, D.
K
H
D
BC
A
RQ
PS
C
D B
A
150
MID-SEGMENT THEOREM
1. In figure, ABCD is an isosceles trapezoid with side lengths AD = BC = 5, AB = 4,
and DC = 10. The point C is on DF and B is the midpoint of the hypotenuse DE in the
right triangle DEF. Then CF =
(Hint: Use mid-segment theorem)
A) 3.25 B) 3.5 C) 3.75 D) 4.0 E)4.25
Answer: D
2. Pre- RMO-2015(15):Let 1
A , 1B , 1
C , 1D be the midpoints of the sides of a convex
quadrilateral ABCD and let 2A , 2
B , 2C , 2
D be the midpoints of the sides of a
convex quadrilateral 1
A1
B1
C1
D . If 2A
2B
2C
2D is a rectangle with sides 4 and 6,
then what is the product of the lengths of the diagonals of ABCD? Answer:208
10
4
55
F
B
D C
A
E
151
(Use Mid-segment theorem)
3. Pre Rmo 2012(10): (Also 10B-2012(14)): ABCD is a square and AB = 1. Equilateral
triangles AYB and CXD are drawn such that X and Y are inside the square. What is
the length of XY?
Answer: 13 . Hint: Find GX, IG = 0.5 unit.
6
4
D2
C2
B2
A2
D1
C1
B1
A1
A
B
C
D
I
G
H
X
Y
B
CD
A
152
4. Pre- RMO (WB)-2015(PII -13): Let ABC be a triangle with base AB. Let D be the
midpoint of AB and P be the midpoint of CD. Extend AB in both directions.
Assuming A to be on the left of B, let X be a point on BA extended further left such
that XA = AD. Similarly, let Y be a point on AB extended further RIGHT such that
BY = BD. Let PX cut AC at Q and PY cut BC at R. Let the sides of ABC be AC =
13, BC = 14 and AB= 15. What is the area of the pentagon PQABR?
Hint: Draw TS, midpoints of XP, QC. Area of AQP, PSQ, CSP are equal. (Problem on mid-
segment and area) Answer: 56
5. Pre- RMO-2013(8): Let AD and BC be the parallel sides of a trapezium ABCD. Let
P and Q be the midpoints of the diagonals AC and BD. If AD = 16 and BC = 20, what
T
S
RQ
P
YX D
C
A B
153
is the length of PQ?(Hint: Use Mid-segment Theorem) Answer: 2
6. Square BCDE and ACD have equal areas. Square BCDE has sides of length 12 cm.
AD intersects BE at F.
Determine the area of quadrilateral BCDF. Answer: 108 sq.cm
7. RMO – 2012: Let ABC be a triangle and D be a point on the segment BC such that is
BDDC 2 . Let E be the midpoint of AC. Let AD and BE intersect in P. Determine
the ratios PE
BP and
PD
AP.
16
20
PQ
B C
A D
F
A
D
EB
C
154
Answer. 1: 1, 3: 1.
y
y
xxx F
P
E
A
BC D
155
AREA OF PARALLELOGRAMS
AND TRIANGLES
1. Definition: A triangular region consists of a triangle and it’s interior. A
circular region consists of a circle and it’s interior. A polygonal region is a
plane figure that is the union of a finite number of coplanar, non
overlapping triangular regions that intersect at vertices or along sides.
Axiom: To every polygonal region there corresponds a unique positive
number, called the area of the region.
Axiom: If two triangles are congruent, then the triangles have equal area.
2. Axiom: The area of a rectangle is equal to the product of the length b of a
base and the corresponding height h. hbA
Theorem: The area A of a square is equal to the square of the length s of a
b
h
C
BA
D
156
side. 2sA .
Area Addition Postulate: The area of the union of two or more non
overlapping polygonal regions is the sum of the areas of the regions.
Theorem: The area of a parallelogram is equal to the product of the
length b of a base and the corresponding height h. hbA
Theorem: The area A of a triangle is equal to one half the product of the
A = s2
s
b
h
BA
CD
157
length b of any side and the corresponding height h. hbA 2
1.
3. Definition: An altitude of a trapezoid is a segment from any point in one
base perpendicular to the line containing the other base. Its length is the
height of the trapezoid.
Theorem: The area A of a trapezoid is equal to one half the product of the
height h and the sum of the lengths of the bases.
b
h
B C
A
h
C
BA
D
158
hbbA
212
1
PROBLEMS
1. PASCAL – 2007(20): In the diagram, rectangle ABCD is divided into two regions,
AEFCD and EBCF, of equal area. If EB = 40, AD = 80 and EF = 30, what is the
length of AE?
A) 20 B) 24 C) 10 D) 15 E) 30
h
b2
b1C
BA
D
C
A
D
BE
F
159
2. Two rectangles overlap with their common region being a smaller rectangle, as
shown. The total area of the shaded region is
A) 45 B) 70 C) 52 D) 79 E) 73
3. A square with perimeter 20 is contained within a larger square of perimeter 28. The
area of the shaded region is
A)8 B) 96 C)24 D)4 E) 28
4
9
2
35
8
160
4. GAUSS-1996(16) : The area of the triangle ABC is
A)32 B) 24 C)16 D)17 E) 20
5. PASCAL-1999(9): In the diagram, each small square is 1 cm by 1 cm. The area of the
shaded region, in square centimeters, is
A)2.75 B) 3 C)3.25 D)4.5 E) 6
6. In the diagram, ABCD is a rectangle with AD =13, DE = 5 and EA = 12. The area of
ABCD is
5
3
62
B
C
A
161
A)39 B) 60 C)52 D)30 E) 25
7. The diagram shows two isosceles right-triangles with sides as marked. What is the
area of the shaded region in square centimeters??
A)4.5 B) 8 C)12.5 D)16 E) 17
8. In the diagram, square ABCD has side length2, with E the midpoint of BC and F the
midpoint of DC. The area of the shaded region BEFD is
A) 1 B) 4/3 C)3/2 D)4 E) 3
D
B C
A
E
3 cm
5 cm
2
F
E
C
BA
D
162
9. Square ABCD has an area of 900. E is the midpoint of AB and F is the midpoint of
AD. What is the area of triangle AEF?
A)100 B) 112.5 C) 150 D) 225 E) 180
10. The base of a triangle is tripled and height is halved. The ratio of the area of the new
triangle to the area of the original triangle is
A) 3 : 4 B) 9 : 4 C) 3 : 2 D) 4 : 3 E) 2 : 3
11. Kalyan cut rectangle R from a sheet of paper. A smaller rectangle is then cut from the
large rectangle R to produce figure S. In comparing R to S
A) the area and perimeter both decrease.
B) The area decreases and the perimeter increases.
C) The area and perimeter both increase.
D) The area increases and the perimeter decreases
E) the area decreases and the perimeter stays the same.
F
E
C
BA
D
5 cm6 cm
8 cm
4 cm
SR
8 cm
8 cm
6 cm
163
12. The length and width of a rectangle are both doubled. When the new rectangle is
compared to the original rectangle
A) the area and the perimeter are the same.
B) The area and the perimeter are both doubled.
C) The area is doubled and the perimeter is four times as large as the
original perimeter.
D) The area is four times as large as the original triangle and the
perimeter is doubled.
E) The area and the perimeter are both four times as large as the original
are and perimeter.
13. In the diagram, square ABCD is made up of 36 squares, each with side length 1. The
area of the square EFGH , in square units, is
A) 12 B) 16 C) 18 D) 20 E)25
14. AMC 8 - 2000(25): The area of a rectangle ABCD is 72 square cm. If point A and the
midpoints of BC and CD are joined to form a triangle AEF, the area of the triangle
AEF is
F
H
G
E
C
BA
D
164
A)21 B) 27 C) 30 D) 36 D)40
15. AMC 8 - 2002(20): The area of triangle ABC is 8 square inches. Points D and E are
the midpoints of AB and AC. Altitude AF bisects BC. The area of the shaded region is
A) 121 B)2 C) 2
21 D)3 E) 3
21
16. AJHSME - 1997(15): Each side of the large square in the figure is trisected (divided
into three equal parts). The corners of an inscribed square at these trisection points, as
shown. The ratio of the inscribed square to the area of the large square is
F
E
D
B
C
A
D E
FB C
A
165
A) 33
B) 95 C)
32 D)
35
E) 97
17. AJHSME -1997(7): The area of the smallest square that will contain a circle of radius
4 is
A) 8 B) 16 C) 32 D)64 E) 128
18. In the diagram ABCD is a square of side 3 units. Points X, G are on side DC with CX
= XG = GD = 1. Points E, Y are on side BC with CY = YE = EB = 1. Line segments
EF and YH are perpendicular to BC and Line segments GF and XH are perpendicular
to DC. The ratio of shaded area to the unshaded area is
B
CD
A
166
A) 2 : 1 B) 7 : 3 C) 7 : 4 D) 5 : 4 E) 3 : 1
19. In the diagram, ABCD is a rectangle. If the area of triangle ABE is 40, then the area of
the shaded region is
A) 20 B) 40 C) 60 D) 50 E) 80
20. 8 – 2011(16): Let A be the area of a triangle with sides of length 25, 25 and 30. Let B
be the area of a triangle with sides of length 25, 25 and 40. What is the relationship
between A and B?
A) BA16
9 B) BA
4
3 C) A = B
D) BA3
4 E) BA
9
16
H
F
X
Y
E
G C
BA
D
10
A
C
B
D E
167
21. 8 – 2011(20): Quadrilateral ABCD is a trapezoid, AD = 15, AB = 50, BC = 20, and
the altitude is 12. What is the area of trapezoid?
A)600 B) 650 C)700 D) 750 E)800
22. 8 – 2003(21): The area of trapezoid ABCD is 164 2cm . The altitude is 8 cm, AB is
10 cm, and CD is 17 cm. What is BC, is centimeters?
A) 9 B) 10 C) 12 D) 15 E)20
23. 8 – 2003(25): In the figure, the area of square WXYZ is 25 2cm . The four smaller
squares have sides 1 cm long, either parallel to or coinciding with the sides of the
large square. In ABC , AB = AC, and when ABC is folded over side BC, point A
coincides with O, the center of the square WXYZ. What is the area of ABC , in
square centimeters?
201215
50
D C
A B
10 8 17
DA
B C
168
A) 4
15 B)
4
21 C)
4
27 D)
2
21 E)
2
27
24. 8 – 2009(7): The triangular plot of land ACD lies between Aspen Road, Brown Road
and a railroad. Main Street runs east and west, and the railroad runs north and south.
The numbers in the diagram indicate distances in miles. The width of the railroad
track can be ignored. How many square miles are in the plot of land ACD?
A)2 B) 3 C) 4.5 D) 6 E)9
25. 8 – 2008(23): In square ABCE, AF = 2 FE and CD = 2 DE. What is the ratio of the
area of BFD to the area of square ABCE?
AO
X
Y
C
Z
W
B
3
3
3
Main
Aspen
Brown C
BA
D
169
A) 6
1 B)
9
2 C)
18
5 D)
3
1 E)
20
7
26. 8 – 2007(23): What is the area of the shaded pinwheel shown in the 55 grid?
A)4 B) 6 C) 8 D) 10 E) 12
27. 8 – 2000(6): Figure ABCD is a square. Inside this square three smaller squares are
drawn with side lengths as labeled. The area of the shaded-shaped region is
D
F
C
BA
E
170
A)7 B) 10 C)12.5 D)14 E) 15
28. 8 – 2005(13): The area of the polygon ABCDEF is 52 with AB = 8, BC = 9 and FA=
5. What is EFDE ?
A) 7 B) 8 C) 9 D) 10 E)11
29. AJHSME – 1993(18): The rectangle shown has length AC =32, width AE = 20, and
B and F are midpoints of AC and AE, respectively. The area of quadrilateral ABDF is
1
1
1
1
3
3
AD
C B
5
9
8
E
C
BA
D
F
171
A) 320 B) 325 C)330 D) 335 E)340
30. The area of rectangle ABCD is 72. If point A and the midpoints of BC and CD are
joined to form a triangle, the area of that triangle is
A) 21 B) 27 C)30 D) 36 E)40
31. AJHSME – 1989(15): The area of the shaded region BEDC is in parallelogram
ABCD is
F
B C
E D
A
D
B
C
A
6
8
10
E
C
A D
B
172
A) 24 B) 48 C) 60 D) 64 E)80
32. In the rectangle shown, the area of the shaded region is
A) 60 2cm B) 20 2cm C) 30 2cm
D) 40 2cm E) 50 2cm
33. 8 - 2009(8): The length of a rectangle is increased by 10% and the width is decreased
by 10%. What percent of the old area is the new area?
A) 90 B) 99 C) 100 D) 101 E) 110
34. Gauss (8th
grade) – 2016 (22): In rectangle ABCD, what is the total area of the
shaded region?
12 cm
2 cm
5 cm
173
A) 25 2cm B) 31 2cm C) 39 2cm
D) 35 2cm E)41 2cm
35. A square and a triangle have equal perimeters. The lengths of the three sides of the
triangle are 6.1 cm, 8.2 cm and 9.7 cm. What is the area of the square in square
centimeters?
A) 24 B) 25 C) 36 D) 48 E) 64
36. A square has sides of length 10, and a circle centered at one of its vertices has radius
10. What is the area of the union of the regions enclosed by the square and the circle?
A) 25200 B) 75100 C) 10075
D) 100100 E) 125100
37. 12A – 2004(8): In the Figure, EAB and ABC are right angles, AB = 4, BC = 6,
AE = 8, and AC and BE intersect at D. What is the difference between the areas of
ADE and BDC ?
5 cm
2 cm
6 cm
4 cm
6 cm
HE
D
B C
A
G
F
174
A)2 B) 4 C) 5 D)8 E) 9
38. 12A – 2003(14):Points K, L, M and N lie in the plane of the square ABCD so that
AKB, BLC, CMD, and DNA are equilateral triangles. If ABCD has an area of 16, find
the area of KLMN.
A) 32 B) 31616 C) 48 D) 31632 E) 64
39. 10B-2012(14): Two equilateral triangles are contained in a square whose side length
is 32 . The bases of these triangles are opposite sides of the square, and their
intersection is rhombus. What is the area of the rhombus?
4
68D
A B
E
C
N L
M
K
A B
D C
175
A) 2
3 B) 3 C) 122 D) 1238 E)
3
34
Hint: See mid-segment theorem Question 3
40. 8 -2013(24): Squares ABCD, EFGH, and GHIJ are equal in area. Points C and D are
midpoints of sides IH and HE, respectively. What is the ratio of the area of the shaded
pentagon AJICB to the sum of the areas of the three squares?
A) 4
1 B)
24
7 C)
3
1 D)
8
3 E)
12
5
I
G
H
X
Y
B
CD
A
BA
C I
D
HE
GF J
176
41. 10A- 2016(11): What is the area of shaded region of the given 5 8 rectangle?
A) 5
34 B) 5 C) 5
4
1 D) 6
2
1 E) 8
Hint: Through the center of the rectangle draw lines parallel to the sides of rectangle.
42. 10A – 2002(25): In trapezoid ABCD with bases AB and CD, we have AB= 52, BC =
12, CD = 39, and DA = 5. The area of ABCD is
A)182 B) 195 C)210
D) 234 E) 260
43. Contest Book VI - Page 35, Q 14: The convex pentagon ABCDE has
120BA , EA =AB = BC = 2 and CD = DE = 4. What is the area of
ABCDE?
7
7
4
4
1
1
1
1
5
39
52
12
D
BA
B
177
A) 10 B) 37 C) 15 D) 39 E) 512
Hint: There are 3 different ways this problem can be solved. 1) Draw EC 2) Extend AB either way
and the given figure is inscribed in an equilateral triangle 3) Produce EA, CB to intersect at P.
44. Squares ABCD and EFGH are congruent, AB = 10, and G is the center of square
ABCD. The area of the region in the plane covered by these squares is
A) 75 B) 100 C) 125 D) 150 E)175
45. Point F is taken in side AD of square ABCD. At C a perpendicular is drawn to CF,
meeting AB extended at E. The area of ABCD is 256 square inches and area of
triangle CEF is 200 square inches. Then the number of inches BE is:
CE
A B
D
E
H F
G
B
CD
A
178
A) 12 B) 14 C) 15 D) 16 E) 20
Hint: Prove triangles DCF and BCE are congruent.
46. Given parallelogram ABCD with E the midpoint of diagonal BD. Point E is connected
to a point F in DA so that DF = 3
1DA. What is the ratio of the area of triangle DFE to
the area of quadrilateral ABEF?
A) 1 : 2 B) 1 : 3 C) 1 : 5 D) 1 : 6 E) 1 : 7
47. PASCAL-2016(23): In the diagram, PQR is isosceles with PQ = PR = 39 and
SQR is equilateral with side length 30. The area of PQS is closest to
EB
CD
A
F
E
C
DA
F
B
179
A) 68 B) 75 C) 50 D) 180 E) 135
ANSWERS
1 D 2 B 3 C 4 D 5 B 6 B 7 B 8 C
9 B 10 C 11 E 12 D 13 D 14 B 15 D 16 B
17 D 18 A 19 B 20 C 21 D 22 B 23 C 24C
25 C 26 B 27 A 28 C 29 A 30 B 31 D 32 E
33 B 34 D 35 C 36 B 37 B 38 D 39 D 40 E
41 D 42 C 43 B 44 E 45 A 46 C 47 B
48. EUCLID 2015(7a): In the diagram, ACDF is a rectangle with AC = 200 and
CD = 50. Also FBD and AEC are congruent triangles which are right-angled at B
and E, respectively. What is the area of the shaded region
39
30
39
S
Q R
P
180
Answer: 2500
Hint: Prove ABEF, BCDE are rectangles. Diagonals of a parallelogram divide into 4 triangles of
equal area.
49. In the diagram, ABCD is a rectangle with points P and Q on AD so that
AB = AP =PQ = QD. Also, point R is on DC with DR = DC. If BC = 24, what is the
area of PQR ?
Answer: 32
50. Trapezoid ABCD has three equal sides AB = AD = DC. The base BC is 2 cm less than
the sum of the lengths of the other three sides. The distance between the parallel sides
is 5 cm. Determine the area of trapezoid ABCD.
ED
A
F
CB
24
R
Q D
B C
A P
181
Answer: 125
51. CAYLEY - 1998(16): Rectangle ABCD has length AB = 9 and width AD = 5.
Diagonal BD is divided into 5 equal parts at E, F, G and H. Determine the area of the
shaded region.
Answer: 18
Hint: All triangles have the same altitude when the bases are on the diagonals DB.
52. Triangle ABC has side BA extended to D so that BA = AD, AC extended to F so that
AC = CF, and CB extended to E so that CB = BE. If the area of DEF = 1176 2cm ,
determine the area of ABC .
D
B C
A
GF
E
H
D
B
C
A
182
Answer: 168 sq.cm. Hint: Prove all 7 triangles have equal area.
Hint: Draw DC, BF, AE. Median divides a triangle into two triangles of equal area.
53. Rectangle DEFG has square ABCD removed leaving an area of 92 2m . Side AE = 4 m
and side CG = 8 m. Determine the area of rectangle DEFG. Answer: 117 sq.m
E
F
D
AC
B
BA
C G
E
D
F
183
54. PQRS is a rectangle. A and B are points on QR such that QA =AB = BR. C is the
midpoint of PQ. The area of ACS is 10 2cm . Determine the area of rectangle PQRS.
Answer: 30 sq.cm
55. In the following slightly irregular shape.
□ AB = 50 cm, CD = 15 cm, EF = 30 cm
□ the area of the shaded triangle DEF is 210 square cm
□ the area of the entire figure ABCDE is 1000square cm.
Determine the length of AE. Answer: 10.9 cm
C
BA R
P
Q
S
184
56. A rectangular plot of land measuring 16 m by 12 m is used as a vegetable garden.
ABCD represents the plot with AB = 16 m and BC = 12 m.
The midpoints of sides AB, BC, CD and AD are P, Q, R and S, respectively. T is the
midpoint of PS.
Peas are planted in the triangular section labeled APS , green beans are planted in
the triangular section labeled QCR , carrots are grown in the quadrilateral section
labelled TSDR, and potatoes are grown in the quadrilateral section labelled PBQT.
The shaded region shown as QRT has nothing planted in it.
What fraction of the area of the entire garden has nothing planted in it?
G
B C
A E
D
F
185
Answer: ¼
Hint: Area of PQRS is one half of ABCD.
Beans
Carrots
PotatoesPeas T
R
Q
P
S
C
A
D
B
186
57. The dots in the diagram are one unit apart, horizontally and vertically. Determine the
area of the figure. Answer: 35
y
x
187
CIRCLES
Circles model repetitive and periodic behavior like day, night, seasons, heartbeat.
1. Definition: A circle is the set of coplanar points at a given distance from a
given point in the plane. The given point is called the center.
2. Definition: A radius of a circle is a segment determined by the center and a
point on the circle.
3. Definition: A chord of a circle is a segment whose endpoints lie on the
circle.
O
FE
188
Theorem: If a line through the center of a circle is perpendicular to a
chord, then it bisects the chord.
Theorem: In the same circle or in congruent circles, two chords have the
same length if and only if they are the same distance from the center(s) of
the circle(s).
Theorem: Equal chords of a circle are equidistant from the center.
Theorem: The perpendicular bisector of a chord contains the center of the
circle.
Theorem: No three different points of a circle are collinear.
AB = CD OE = OF
D
C
A B
O
E
F
189
4. Definition: A diameter of a circle is a chord that contains the center of the
circle.
5. Definition: The interior of a circle is the set of points in a plane whose
distance from the center is less than the radius. The exterior of a circle is
the set of points in a plane whose distance from the center is greater than
the radius.
B
O
A
Circle Interior
190
6. Definition: Two or more circles are said to be concentric if they have the
same center.
7. Definition: Congruent circles are circles with equal radii.
8. Definition: A circle is circumscribed about a polygon when the vertices of
the polygon lie on the circle. The polygon is inscribed in the circle.
O
191
9. Definition: Concurrent lines are two or more lines that intersect in a single
point. The point is called the point of concurrency.
Definition: The circumcenter of a triangle is the point of concurrency of the
perpendicular bisectors of the sides of the triangle. The circumcircle is the
circumscribed circle.
k
mn
192
Theorem: The perpendicular bisector of the sides of a triangle are
concurrent in a point equidistant from the vertices of the triangle.
Theorem: A circle can be circumscribed about any triangle.
10. Definition: A central angle of a circle is an angle that lies in the plane of
the circle and whose vertex is the center of the circle.
CIRCUMCENTER
B C
A
193
A minor arc of a circle is the set of points on the circle that lie on or in the
interior of a central angle.
Intercepted arc.
Major arc BA
11. Definition: A semicircle is the set of points which includes the endpoints of
a diameter and the points of the circle in a given half plane formed by the
line containing the diameter.
AB
OAB
194
12. Definition: The degree measure of a minor arc is the measure of its
central angle.
Axiom: If P is on arc AB, then
APBmBPmAPm .
Theorem: In the same circle or in congruent circles, two minor arcs have
equal measures if and only if their chords have the same length.
13. Definition: The degree measure of a major arc is 360 minus the measure
of the corresponding minor arc.
Definition: The degree measure of a semicircle is 180.
Definition: An angle inscribed in a circle is an angle whose vertex is on the
circle and whose sides contain chords of the circle.
Theorem: The measure of an inscribed angle is one half the measure of its
intercepted arc. Euclid Book III, Proposition 20 Reference: G. Polya
O
C
A
B
195
Proof: If the angle at the center is given, the angle at the circumference is not yet
determined, but can have various positions.
The special case in which one of the sides of the angle at the circumference passes
through the center of the circle is “leading”.
β
α
α = 2β
O
P
BA
α
β
α = 2β
Leading case
A
O
P
B
196
From two such special angles, we can combine the general angle at the
circumference by addition or subtraction
Theorem: If two inscribed angles intercept the same arc or arcs of equal
measure, then the angles are equal.
APC = β'
BPA = β
AOC = α'
AOB = α
β - β'
α - α'
β
α
General case
α'
β'
α -α'= 2(β - β')α +α'= 2(β+β')
A
OO
P
BA
P
B
C
A B
CD
197
Theorem: Angles in the same segment of a circle are equal.
Theorem: If an angle is inscribed in a semi circle, then it is a right angle.
Theorem: Two inscribed angles that intercept the same arc or congruent
arcs are congruent.
Theorem: If a quadrilateral is inscribed in a circle, then the opposite
angles are supplementary.
14. Definition: Two arcs are said to be congruent if they have the same
measure and lie in the same or in congruent circles.
BOA
P
A + C = 180°
O
C
B
D
A
198
Theorem: In the same circle or in congruent circles, two minor arcs have
equal measures if and only if their chords have the same length.
AB CD
D
C
BA
199
PROBLEMS
1. 8 -2014(15): The circumference of the circle with center O is divided into 12 equal
arcs, marked the letters A through L as seen below. What is the number of degrees in
the sum of the angles x and y? Answer: x =30o, y = 60
o
A) 75 B)80 C)90 D) 120 E) 150
2. AMC Book I, Page 46, Q 35: A rhombus is formed by two radii and two chords of a
circle whose radius is 16 feet. The area of the rhombus in square feet is:
y
x
F
I
J
K
A
L D
H
BC
G
O E
200
A) 128 B) 128 3 C) 256 D) 512 E) 512 3
3. Points A and B are on a circle of radius 5 and AB = 6. Point C is the midpoint of the
minor arc AB. What is the length of the line segment AC
A) 10 B) 2
7 C) 14 D) 15 E) 4
4. AMC Book VI, Page 44, Q 18: Triangle ABC is inscribed in a circle, and B =
C = 4 A . If B and C are adjacent vertices of a regular polygon of n sides
inscribed in this circle, then n =
16
16
16
5 5
3
C
O
BA
201
A) 5 B) 7 C) 9 D) 15 E)18
ANSWERS
1 C 2 B 3 A 4 C
5. In the diagram, OC is a radius of the larger circle and a diameter of the smaller circle
which has B as center. Prove that CD = DE.
6. O is the center of a circle with points A, B, C, and D on the circumference. If
50BOC and CO = CD, determine the measure of BAD . Answer: 55o
A
B C
DB
O
C
E
202
7. A circle with center O has points A, B and C on its circumference. 20OBA and
52OCA . Determine the measure of BOC and the measure of BAC . What
is the relationship between these two angles? Answer: 64o, 32
o
8. A circle with center O is drawn around OBD so that B and D lie on the
circumference of the circle. BO is extended to A on the circle. Chord AC intersects
OD and BD at F and E, respectively.
50°
O
A
DB
C
52°20°
O
C
A
B
203
If 19BAC and 99OFA , determine the measure of BEC . Answer: x =50o
9. A is the center of a circle which passes through B. B is the center of a circle which
passes through A. CAE and CBF are straight line segments. If 78F , determine
the measure of C . Answer: x =8o
10. A circle with center O is drawn with points P, Q and S on the circumference such that
PQ = PS = 12 m. PO is extended to meet QS at R such that QSPR and OR = 1 m.
Determine the radius of the circle. Answer: 8 m
x°
99°19°
E
F
BO
A
CD
78°
E
F
A B
C
204
11. Pre - RMO 2012(14): O and I are the circumcenter and in-center of ABC
respectively. Suppose O lies in the interior of ABC and I lies on the circle passing
through B, O and C. What is the magnitude of BAC in degrees? Answer: 60
1 m
12 m
12 m
R
Q
OP
S
I O
B C
A
205
12. Pre - RMO 2014(16): In a triangle ABC, let I denote the incenter. Let the lines AI, BI
and CI intersect the incircle at P, Q and R, respectively. If 40BAC , what is the
value of QPR in degrees? Answer: 55
13. RMO – 2007 (5): A trapezium ABCD, in which AB is parallel to CD, is inscribed in a
circle with center O. Suppose the diagonals AC and BD of the trapezium intersect at
M, and OM = 2. If 60AMB , find, with proof, the difference between the lengths
of parallel sides.
RQ
P
I
A
B C
206
Answer: 32 ba Hint: Triangles MDC and ABM are equilateral. BD = a + b, MD =MN + DN
14. RMO – 2007 (5): A trapezium ABCD, in which AB is parallel to CD, is inscribed in a
circle with center O. Suppose the diagonals AC and BD of the trapezium intersect at
M, and OM = 2. If 60AMD , find, with proof, the difference between the lengths
of parallel sides.
N
AB
M
O
C D
207
Answer: 32 ba
15. RMO – 2001 (1): Let BE and CF be the altitudes of an acute triangle ABC, with E on
AC and F on AB. Let O be the point of intersection of BE and CF. Take any line KL
through O with K on AB and L on AC. Suppose M and N are located on BE and CF
respectively, such that KM is perpendicular to BE and LN is perpendicular to CF.
Prove that FM is parallel to EN.
M
B
O
D C
A
208
Hint: Use properties of cyclic quadrilateral.
16. RMO – 2013 MUMBAI (1): Let ABC be an isosceles triangle with AB = AC and let
C denote its circumcircle. A point D is on arc AB of C not containing C. A point E
is on arc AC of C not containing B. If AD = CE prove that BE is parallel to AD.
θ
N
M
K O
F
E
B C
A
L
E
A
B C
D
209
17. RMO – 92(4): ABCD is a cyclic quadrilateral with AC BD; AC meets BD at E.
Prove that 22222 4RDECEBEAE , where R is the radius of the
circumscribing circle.
Hint: 2)(2)(22 EPCPEPAPCEAE
18. ABCD is a rectangle inscribed in a quarter-circle as shown. A is on BF, B is the center
of the quarter-circle, C is on BE, and D is on arc FE.
Q
P
B
D
O
A
C
E
210
If AD = 12 cm and CE= 1 cm, determine the length of AF. ? Answer: 8 cm
19. Pre RMO (Mumbai) 2015(16): In an acute-angled triangle ABC, let D be the foot
of the altitude from A, and E be the midpoint of BC. Let F be the midpoint of AC.
Suppose 40BAE . If DFEDAE , what is the magnitude of ADF in
degrees? Answer: 40o. Hint: Prove ADEF is cyclic.
AD
F
B EC
F
D EB C
A
211
Acknowledgement:
Great Moments in Mathematics, Howard Eves, MAA
EUCLID’S 5 POSTULATES (Elements, Book –I)
1. A straight line segment may be drawn connecting any two given points.
2. A straight line may be produced continuously in a straight line in either
direction.
3. A circle may be drawn with any given point as the center and passing
through any given second point.
A B
lA B
O
P
212
4. All right angles are equal to one another.
5. Euclid’s 5th
Postulate (or Parallel Postulate):
The most famous single utterance in the history of science.
If a straight line falling on two straight lines makes the interior angles on
the same side together less than two right angles, the two straight lines, if
produced indefinitely, meet on that side on which the angles are together
less than two right angles.
DEF = 90°ABC = 90°
B E
C
F
DA
l
m
n
2
1
1 + 2 < 180°
213
Alternate forms of Euclid’s 5th
Postulate:
i) There exists a pair of coplanar straight lines everywhere equally distant
from one another.
ii) Through any point not on a given line one and only one line can be
drawn parallel to the given line. Playfair Axiom (1748- 1819)
iii) There exists a pair of similar non-congruent triangles.
iv) If in a quadrilateral a pair of opposite sides is equal and if the angles
adjacent to a third side are right angles, then the other two angles are
also right angles.
v) If in a quadrilateral three angles are right angles, then the fourth angle
is also a right angle.
vi) There exists at least one triangle having the sum of its three angles
equal to two right angles.
vii) Through a point within an angle less than 60 there can always be
drawn a straight line intersecting both sides of the triangle.
viii) A circle can be passed through any three non-collinear points.
ix) There is no upper limit to the area of a triangle.
6. NON-EUCLEADEAN GEOMETRY
i) Elliptic Geometry (or Riemannian geometry):
In this geometry straight line is not infinite but finite and closed. In this
geometry parallel postulate and axioms of betweenness are not true.
Example: Great circles on a sphere are straight lines and every two straight
lines intersect and from an external point no line can be drawn parallel to a
214
given straight line.
Note: l amd m are two lines. X is any point outside the line m. No line can be drawn through X
parallel to m. Line l is intersecting m at points A and B.
ii) Hyperbolic Geometry:
Plane consists of the points interior to the circle and points outside the
circle are disregarded. Each point inside the circle is called a non-Euclidean
point and each chord of the circle is called a non-Euclidean straight line. In
this geometry through a point not on the line infinitely many straight lines
can be drawn having no point in common with the given line.
l
mX
B
A
215
Note: l is a line and P is any point outside the line l. There are many lines through P, not intersecting
l. Lines m and n are parallel to l.
m
n
l
P
216
SOLIDS
1. Definition: A polyhedron is a figure formed by four or more noncoplanar
polygonal regions which enclose a part of space.
Compare: Polygon and polyhedron.
Polygon is a figure in a plane enclosed by line segments. Polyhedron is a
solid bounded by polygons.
Note: There exists only 5 regular polyhedra (called Platonic solids).
Tetrahedron (4 triangular faces) Cube (6 square faces)
Octahedron (8 triangular faces) Dodecahedron (12 pentagonal faces)
217
Icosahedron (20 triangular faces)
Axiom: To every solid polyhedron there corresponds a unique positive
real number called the volume of the polyhedron.
Definition: A prism is a polyhedron with two congruent faces contained in
parallel planes. All other faces are parallelograms. Parallel planes are
called lateral faces.
Axiom: Volume of a prism: hbasetheofareaV ) (
Theorem: The lateral edges of a prism are equal and parallel.
218
Theorem: The bases of a prism have equal areas.
Theorem: Diagonal of a cuboid = 222 hwl
Axiom: The volume of a rectangular solid is hblV
Cavalier’s principle
Definition: Given a plane and two solids, if every plane parallel to this
plane intersects one of the solids also intersects the other so that each
l
h
w
d
219
pair of cross sections formed have the same areas, then the solids have
the same volume.
2. Theorem: All cross sections of a prism have equal area.
3. Definition: The lateral area of a prism is the sum of the area of its lateral
faces. The total area of a prism is the sum of its lateral area and the areas
of its bases.
Theorem: Lateral area of a Prism= pl , p is the perimeter of the base, l is
the length of the lateral edge.
Theorem: Lateral area of a right prism = ph
4. Definition: A cross section of a prism is a section parallel to the bases of
the prism.
5. Definition: A right section of a prism is a section perpendicular to the
lateral edges of the prism.
r
h
220
6. Definition: A pyramid is a polyhedron formed by a polygonal region in a
plane R, a point P not in the plane R, and the triangular regions formed by
joining the vertices of the polygonal region with P.
A square pyramid
Definition: A regular pyramid has a regular polygon for a base, and its
altitude passes through the center of the base. The slant height of a
regular pyramid is the height of any lateral face.
Theorem: The lateral faces of a regular pyramid are congruent isosceles
triangles.
Theorem: Lateral area of a Regular Pyramid =2
pl, where l is the slant
height, p is the perimeter of the base.
Theorem: Volume of a pyramid =3
hB, where B is the area of the base, h
221
is the altitude.
PROBLEMS
1. The surface area of a cube is 96 2cm .The volume of the cube, in 3cm is
A)16 B) 64 C)8 D) 512 E) 216
2. A cube has a volume of 125 3cm . What is the area of one face of the cube?
A) 20 2cm B) 25 2cm C) 413
2 2cm D) 5 2cm E) 75 2cm
3. The base of a rectangular box measures 2 cm by 5 cm. The volume of the box is 30
3cm . What is the height of the box?
A) 1 cm B) 2 cm C) 3 cm D) 4 cm E) 5 cm
4. One thousand unit cubes are fastened together to form a large cube with edge 10 units;
this is painted and then separated into the original cubes. The number of these unit
cubes which have at least one face painted is
A) 600 B) 520 C) 488 D) 480 E) 400
h
r
222
5. 10A- 2016(5): A rectangular box has integer side lengths in the ratio
1: 3: 4. Which of the following could be the volume of the box?
A)48 B) 56 C) 64 D) 96 E) 144
6. 10A – 2013(14): A solid cube of side length 1 is removed from each corner of a solid
cube of side length 3. How many edges does the remaining solid have?
A) 36 B) 60 C) 72 D) 84 E)108
7. 10A – 2003(3): A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by
removing a cube 3 cm on a side from each corner of this box. What percent of the
original volume is removed?
A) 4.5 B) 9 C) 12 D)18 E) 24
8. 10A – 1998(18): A right circular cone of volume A, a right circular cylinder of
volume M, and a sphere of volume C all have the same radius, and the common
height of the cone and cylinder is equal to the diameter of the sphere. Then
A) 0 CMA B) CMA C) CMA 2
D) 0222 CMA E) CMA 322
9. A company sells peanut butter in cylindrical jars. Marketing research suggests that
using wider jar will increase sales. If the diameter of the jars is increased by 25%
without altering the volume, by what percent must the height be decreased?
A) 10 B) 25 C) 36 D) 50 E) 60
10. 12A -05(22): A rectangle box P is inscribed in a sphere of radius r. The surface area is
384, and the sum of the lengths of its 12 edges is 112. What is r?
A) 8 B) 10 C) 12 D) 14 E) 16
ANSWERS
223
1 B 2 B 3 C 4 C 5 D
6 D 7 D 8 A 9 C 10 B
11. A rectangular box has dimensions 9 cm by 6 cm by 24 cm. A second rectangular box
has volume one-half of the first and has a base 6 cm by 4 cm. What is the height of the
second box? Answer: h = 27
12. A square piece of cardboard has an area of 36 square cm. A square of 1 square cm. is
cut from each corner. The sides are folded in order to make an open box. What is the
volume of the box? Answer: 16 cm3
13. The side, front and top faces of a cuboid have areas of x2 , 2
y and xy25 square cm,
respectively. Determine the volume of the cuboid in terms of x and y. Answer: 5xy
14. A cube has surface area of 24 square cm. What is the volume of the box?. Answer: 8
15. Shanghai Exam: A painted wooden cube with a 9 cm edge is cut up into little cubes
with a 3 cm edge. There are thus 27 such little cubes. How many of them will have
only 2 painted sides?
Answer: 3 faces = 8, 2 faces = 12, 1 face = 6, no faces = 1
16. A wooden cube is cut into n cubes each of edge length 1 unit. The combined surface
area of the n cubes is eight times the surface area of the original uncut cube.
Determine the edge length of the original uncut cube. Answer: 8 units
17. POTWE – 2014 – 29: Ship All is a company that ships medium-sized boxes. The
company will only ship boxes where the sum of the length, width and height is at least
100 cm and at most 1000 cm. If the length, width and height of the boxes are in the
ratio 4: 3: 5 and the volume is less than 2 cubic meters.
Determine the dimensions of the smallest and the largest boxes which can be shipped.
Answer: Smallest box 36, 27, 45, Largest box 128, 96, 160 cm
224
18. EUCLID – 2017(3c): One of the faces of a rectangular prism has area 27 2cm .
Another face has an area 322cm . If the volume of the prism is 144
3cm , determine
the surface area of the prism in2cm . Answer: 166 sq.cm
