This Tutorial Will Cover Everything You Need to Know About Geometry for the GMAT

This tutorial will cover everything you need to know about geometry for the GMAT, from angles and lines through triangles, quadrilaterals and circles to Cartesian geometry and 3D geometry.

None of theses concepts is difficult and you will pleased to learn that trigonometry is NOT required in the GMAT.

How well you do on geometry questions will depend mainly on how good you are at remembering everything you learn in this tutorial.

There is really only one very important tip that I want to give you for geometry, draw your own diagrams. This is especially important with data sufficiency questions since the diagrams provided in the GMAT for these questions can be misleading.

When drawing diagrams the following guidelines are useful.

In the diagrams in the GMAT everything that looks like a straight line should be considered a straight line and where lines appear to touch they do but the scale

of the drawings may not be accurate, i.e. just because two lines look the same length does not mean they are the same length, just because one angle looks larger than another does not mean that it is. Lines which are parallel or perpendicular or the same length will be marked as such on the diagram or described in the accompanying text.

If you know the lengths of lines or the size of angles try to make the diagram more or less to scale. This should reduce the number of mistakes you make since you can see what is going on more clearly.

Unless you know lines or angles are definitely equal then draw them as different as you can. This will stop you making wrong assumptions when you are answering the question.

If you are answering questions from a book then you should still draw your own diagram and avoid just adding notes to an existing diagram because in the real test all diagrams will be on a computer screen.

Line is long thin Mark which is drawn or painted on a surface.

A ‘A’ is a line

x y

A D B

C

xy

1800

A

Liner angle

Liner angle is 1800

x + y = 1800

x and y are supplementary angle for each other.

The line CD perpendicular is perpendicular to line AD

Perpendicular always 900 angle. <BDC = 900 and <ADC = 900

900 angle is Right angle

x + y = 900

x and y are Complementary for each other.

a

x y

z

b c

Some lines intersect each other and form some angles. In the figure below:

Opposite angles are always equal.

<x is opposite angle <c. x = c

x = c x + y = b + c a + x + y

b = y a + b = y + z = b + c + z

a = z x + a = z + c

x + y + z = 1800, y + z + c = 1800, b + c + z = 1800, a + b + c + 1800, a + b + x =

180, x + y + z + a + b + c = 3600

∴A circle is 3600

All angles in the GMAT are measured in degrees and there are 360° in a full circle.

Crossing lines

When two lines cross they form two pairs of angles. The angles opposite each other are equal and angles next to each other add up to a straight line, i.e. 180°.

Therefore, in the diagram above, the two angles marked x are equal, the two angles marked y are equal and finally x + y = 180°.

Parallel lines

Parallel lines, on a plane, are those which never cross. They are always the same distance apart however far you extend them. The symbol < is often used to show that two lines are parallel.

If a line crosses two parallel lines then it will form two sets of four equal angles, see the diagram below.

All angles marked x are equal and all angles marked y are equal.

D

b s s b A

b s bs

B

s b s b

C

E

Perpendicular lines

Lines which meet at right angles, i.e. 90°, are perpendicular lines.

I leave it as an excercise for you to work out why all four angles in this diagram are right angles (using what you have already learned about crossing lines).

When some parallel lines intersect by a line or more then form some angles. Determine its by

small angles and big angles.

∴ In the figure above, A,B and C parallel lines are intersected by DE .

∴ b = b = b = b = b = b

s = s = s = s = s = s

b + s = 1800

a b

c

h g

L1

L2

L3f

d

e

A D

a

C B

L1

L2

L3

450

350

Problems on above discussion:

(1) If cyclist has traveled 2000 around a circular track what fractional part of track has he driven?

(2)

In the figure above, In L1 L2 L3, then which of the following angles must be equivalent?

A. a and b

B. g and f

C. h and b

D. a and d

E. f and d

3. In the figure above. In L1 L2 L3. What is the value of a?

A. 70

B. 100

C. 80

D. 140

E. 150

(x+40)0

L 60(x-30)0 L1

L2

150y

x

M+100

x+30

4. In the figure above. If, L1 L2. What is the value, x = ?

A. 30

B. 50

C. 550

D. 80

E. In can not be deter mined from the information given.

5. In the figure above x+y?

A. 30

B. 50

C. 600

D. 650

E. It can’t be determined from the information given.

Triangle Quadrilateral Polygon/Pentagon

a

s c

a s

c d

A two- dimensional geometric figure formed of three or more straight sides is polygon.

There are some kinds of polygon-

Sides Name

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

The sum of interior angles = (s-2) × 180

s = the number of sides

= a + b + c = 180

1 × 180 = 1800

a + b + c + d = 3600

2 × 180 = 3600

a b e

c d

x

3 ×180 = 5400

Regular Polygon: All sides of any polygon are equal.

Single angle of any regular Polygon =

( S−2)×1800

S

Regular Polygon

Single angle × S = (S-2) × 1800

S = the number of sides.

What kind of Regular Polygon it is.

Triangle Inequality

The triangle inequality says that no one side of a triangle can be longer than or equal to the sum of the other two.

The reason for this is fairly obvious when you think about it because if one side is longer than or equal to the sum of the other two then you cannot join the two shorter sides to form a triangle.

Therefore, if we have a triangle with sides of length r, s and t.

600

600 600

a

a

aa a

a/2 a/2

a

300 300

TRIANGLE

We can write down the following inequalities.

TRIANGLE

AREA=b×h2

Perimeter= 3a

a a

b/2 b/2

b

Isosceles

a a

Isosceles

y

x x

h

b

area=b×h2

b = base, h = height

Perimeter = 2 a + b

b/lHypotenuse

b/l

Right angle Triangle

Right angle Triangle is very important .Please do not memories the formulas rather than work

with the problems and try to perceive.

area=b×L2

b = base

L = Lage

H = Hypotenuse

Pythagorean theorem: b2 + L2 = (hypo)2

∴hypo2 = 32 + 42

h2 = 9 + 16 = 25

h=√25= 5

3?

4

RIGHT ANGLE TRIANGLE

15 39

? = 36

49 ? = 175

168

Please memorize below figures:

3 : 4 : 5

6 : 8 : 10

5 : 12 : 13

7 : 24 : 25

9 : 40 : 41

11 : 60 : 61

12 : 35 : 37

∴ 5 : 12 : 13

5 × 3 ? 13 × 3

12 × 3

7 : 24 : 25

7 × 7 : 24 × 7 : ?

25 × 7

49 : 168 : 175

45 25

45

30

600

30

60

a

b c

area = ?

45 : 45 : 90

1 : 1 : √2

x : x : x √2x/√2 : x/√2 : x

45 : 45 : 90

25

√2:25

√2:25

∴ area =

25

√2×25

√22

=2254

lowest

medium Largest

10√320

10

60

3

15

15

3

30

30

6030

15

15√3

30 : 60 : 90

x : x√3 : 2x

x2 :

x√32 : x

x

√3 : x :

2x

√3

x

300

x√3 2x 600 900

area=15×15√32

=225√32

A Note on Square Roots

You will need to know about simplifying square roots if you are to use the Pythagoras theorem successfully in the GMAT.

Basically if you are taking the square root of a number which has a factor which is a perfect square (like 4, 9, 16, 25....) then you can take the factor out of the square root.

For example, if you have a right angled triangle with short sides 4 and 2 what is the length of the hypotenuse?

To solve this you will use Pythagoras. Set c to be the length of the hypotenuse.

You will never find amongst the answers in a GMAT question because 4 is a factor of 20 and is also a perfect square and so we can simplify this square root.

...and this is the answer that you would find in a GMAT question.

Whenever you have a root in your answer you need to make sure that it is in its simplest form which means taking any factors which are perfect squares out of the root.

A

B C

x

y z

D

E F

x

y z

A

B C

D E

8

6

8

SIMILLAR TRIANGLE

In the figure Δ ABC and Δ DEF

<A = <D, <B = <E, <C = <F

Δ ABC I Δ DEF are similar triangle. The ratio of the sides of triangles ia equal.

ABDE

= ACDF

=BCEF

Or,

ABBC

=DEDF

= ABAC

=DEDF

ACBC

=DFEF

DE || BC, BD = ?

Δ ABC and Δ ADE Similar

A

B C

D E

3x

12

?

2x

We know that..

BCDE

= ABAD

=86= AD+BD

8

=86=8+BD

8

= 48 + 6BD = 64

= 6BD = 64-48

BD=166

=83

DE || BC

BC = ?

3 x3x+2 x

=12BC

3x5x

=12BC

3BC = 60,BC = 20

NW

N

NE

W E

SW

S

SE

450450 450

450450

450450450

A

F

B CE

A

C B C

Some important discussion:

4 (AF2 + CF2) = 5 AC2

The angle between a side of a triangle (AC) and an extension of another side (BC) is called

exterior angle of a triangle.

Here the angle marked as q is called the exterior angle.

Further ÐC + Ðq = 180°

Ðq = 180° - ÐC.

A

F E O

B C D

A A’

B C B’ C’

AA’

B C B’ C’ A A’

B C B’ C’

A straight line drawn from a vertex of a triangle

perpendicular to the opposite side is called altitude.

There are three altitudes, in a triangle The point of intersection of

altitudes is called orthocentre.

Monster diagram

Four important postulates on congruent triangles.

1. SSS Rule (side – side – side):Two triangles are congruent if three sides of

one triangle are equal to the corresponding sides of the other triangle.Here AB = A’B’; BC = B’C’ ; CA = C’A’.Then ABC = A’B’C’

2. SAS Rule (side – angle – side):If any two sides and the angle included

between them of one triangle is equal to another, then the two triangles are congruent.Here AB = A’B’ ; AC = A’C’ and ÐA = ÐA’

3. ASA Rule: (angle – side – angle):If in two triangles any two angles and two corresponding

sides are equal, then they are said to be congruent.Here ÐA = ÐA’ ; ÐB = ÐB’ and AC = A’C’.

A A’

BC B’ C’

A

D

B C

4. RHS rule (Right angled triangle – Hypotenuse – side)If the hypotenuse and one side of two right triangles

are equal, then they are congruent.Here AB = A’B’ and AC = A’C’.

In a right angled triangle, a perpendicular drawn from the vertex to the opposite side, divides the given triangle into two similar triangles.BD is perpendicular to AC . ABD and BDC are similar.

A

B C

45 30

45 60

D E

AB AC BF CF BD CD COSquare Δ

area ABF

Δ area

BCDE

Δ area

BOD

2

2

2

2

2

2

2

2

2

2

A quadrilateral is a plane figure bounded by four straight line.

The straight line which joins opposite angular points in a quadrilateral is called a diagonal

The most common quadrilaterals that you will see in the GMAT are the parallelogram, the rhombus, the rectangle and the square.

Parallelogram

A parallelogram is the quadrilateral formed by 2 pairs of parallel sides (thus the name).

Properties

2 pairs of parallel sides. 2 pairs of equal sides. Diagonals bisect each other i.e. cut each other exactly in half. Opposite angles are equal.

Diagonal

Quadrilateral

Rhombus

A rhombus is a parallelogram in which all the sides are equal.

Properties

2 pairs of parallel sides. 4 equal sides. Diagonals bisect each other i.e. cut each other exactly in half. Diagonals are perpendicular. Opposite angles are equal.

Rectangle

A rectangle is a parallelogram where all the angles are right angles (90°).

Properties

2 pairs of parallel sides. 2 pairs of equal sides. Diagonals bisect each other i.e. cut each other exactly in half. Diagonals are equal. All angles are 90°.

Square

A square is a special kind of parallelogram (and rhombus and rectangle) where all the sides are equal and all the angles are right angles.

Properties

2 pairs of parallel sides. 4 equal sides. Diagonals bisect each other i.e. cut each other exactly in half. Diagonals are perpendicular. Diagonals are equal. All angles are equal.

Area

The area of all the previous quadrilaterals, i.e the parallelogram, rhombus, rectangle and square is calculated in exactly the same way; Area = base x height.

Note: In all cases height is measured perpendicularly from the base so in the cases of a rhombus and a parallelogram it is not the same as the length of the side. See diagram below.

A parallelogram is a quadrilateral whose opposite sides are parallel. The figure in the following.

A Rectangle is a parallelogram which one of its angles a right angle. The figure in the following.

A Square is a Rectangle which every of its sides are equal. The figure in the following.

A Rhombus is a square but its angles are not right angles.

Rhombus

Rectangles

Square

A

C

B

D

Flowchart

ABCD is a parallelogram which area s 40. So the area Δ COD

404

=10

x x

ParallelogramA

Rectangle B

C Ex x D

A B

C Ex x D

AB

Δ area = Δ×base

2

(i)

Parallelogram

ABCD area = 40

E is the midpoint of CD

What is area of Δ ABE

Δ =

40×12

2

=402

=10

(ii)

ABCD parallelogram, area of ΔBCE=15

What is area of parallelogram?

Δ area =

×23

2

15 =

2 6

2 = 90

= 45

Δ area =

× base on which triangle is formed 2

The last quadrilateral you will meet in the GMAT is the trapezium. A trapezium is a quadrilateral with 1 pair of parallel sides.

All you need to know about a trapezium is how to calculate its area which is equal to the average of the two parallel sides x perpendicular distance between them

W b

L

bw

L

b w

L

∴The road area of any rectangular figure.

2b(L+W±2b)

b = breadth of road

L = Length of the field

W = width of the field

(+) when road field

A

B C

D E

O

B

A

C C

x x

y

y

O

Δ ABC ,<A = 2 <BOC-1800, <BOC = 900+ 1

2 <A <BOC = 90−1

2 <A x

First some 'circle' vocabulary.

CircumferenceThe edge of a circle.

DiameterA line which joins two points on the circumference of the circle passing through the center of the circle.

RadiusA line which joins the center of the circle to the circumference.

ChordA line which joins two points on the circumference of the circle.

TangentA line which touches the circumference at only one point. A tangent is always perpendicular to a radius or diameter which touches the circumference in the same place.

Area

The area of a circle is PI times the radius squared.

You will not usually need to know the value of PI but just in case it does come up a test you should be aware that it is a little bit more than 3. A good decimal approximation is 3.1 or 3.14 and the fraction 22/7 also gives a good approximation.

Circumference

The length of the circumference of a circle is 2 times PI times the radius.

GMAT students commonly mix up these formulas, if you find you are using the wrong formula try to remember that area comes in square units (meters squared, m2, or centimeters squared, cm2) and so it is the area formula which has radius squared in it.

Sectors and arcs

A sector is like a slice of pizza, it is just a part of circle cut out by two radii. An arc is part of the circumference of the circle.

Length of an arc

If you are asked to calculate the length of an arc you should work out the length of the whole circumference and then take the appropriate fraction of that, remembering that there are 360° in a complete circle.

For example. If we were to work out the length of an arc with radius 5 and an angle of 60° then we would first work out the circumference of the entire circle and then multiply by the fraction of the circle that the sector covers (60° out of 360°).

Area of a sector

Similarly if you are asked to calculate the area of a sector you should work out the area of the whole circle and then take the appropriate fraction of that, remembering that there are 360° in a complete circle.

For example. If we were to work out the area of the sector with radius 6 and an angle of 120° which is shown in the diagram above then we would first work out the area of the entire circle and then multiply by the fraction of the circle that the sector covers (120° out of 360°).

.

area

Sector

Cord

diameter

tangent

radius Circle Center of Circle

. Diameter

A circle is a plane figure contained by a line traced out by a point which moves so that its

distance from a entrain fined point is always the same.

The fixed point is called the center And the bounding lines is called the circumference.

A diameter of a circle is a straight line drawn through the center and terminated both ways by the

circumference.

Circumference.

Center

A B

A BD

O

xII

xII

xII x

II

d = radius + radius = d = 2r

∴ The circumference of circle ×diameter

A chord of a circle is a straight line loaning any two point on the circumference.

Δ OAD≃¿ ¿Δ ODB

A

B C

xo

2x

are

Semicircle

90 90 90

Center angle vs circular angle

Subtended angle

2 <BAC = <BOC

or = <BAC =

12 <BOC

A semi circle is a figure bounded by a diameter of a circle.

Semi circle is the half of the circle.

any angle in a semicircle is a right.

Semi circle Right angle 900|

Circle quadrilateral a pair of opposite angles 1800 or two right angle. In the figure blew.

x

ya

b

A B

C D

Sector area

x + y = 1800

a + b = 1800

In the figure blew-

AC = BD

Sector area

are length

.

.s

s

r

s

Equilateral

Formula

Sector area = = N

360×¿ ¿

r2

Sector arc length = = N

360×¿ ¿

2 r

Side = radius ¿√3

s = r×√3 = r √3

Side = radius 2√3

In the following figure.

s = r × 2√3

N

9090

90

The area of road of a circular garden. In the figure following.

The area of road = B (R + r)

Formula =

11m−60H2 [M = Minute, H = Hour] [ sign not factor]

Clock formula

Example :

6.12 A.M

11. 12−60 . 62

132−3602

=

−2282

∴ Extortion angle 360 – 114 = 2460

12.06 P.M

11×6−60×122

=

66−7202

=

−6542

- 327

∴ interior angle = (360 – 327) = 300

- 114

327

III

II

IV

I4

3

2

1-4 -3 -2 -1

-1

-2

-3

-4

1 2 3 4O X

Y

4

3

2

1-4 -3 -2 -1

-1

-2

-3

-4

1 2 3 4O

P

.

.Q

Co-Ordinate Geometry

Co-ordinate Geometry

3

2

1-4 -3 -2 -1

-1

-2

-3

1 2 3 4O

x1, y1

x2, y2

Y

X

3

2

1-4 -3 -2 -1

-1

-2

-3

1 2 3 4O

x1, y1

x2, y2

Y

X

A

C

1. Distance between two points

2. Midpoint between two points

3. Slope between two points

4. Others between two points

Distance Formula : point distance formula apply

d=√( x2−x1 )2+( y2− y1 )

2

d=√(4−0 )2+(0−3 )2

d=√42+(−3)2

d=√16+9

d=5

point distance Pythagorean theorem

32

1-4 -3 -2 -1

-1

-2

-3

1 2 3 4O

x1, y1

x2, y2

Y

X

A

C

..

.

-1, 3

AC Gi distance Right angle triangle (x,y) coordinates (O,O). AB = 3 Ges BC = 4

∴ AC=√32+42

= 5

Midpoint formula

M=x1+x2

2,y1+ y2

2

Slope

y = mx + b or

x = slope of the line

For example:

Coordinate geometry

Every point in coordinate geometry is specified by two coordinates, an x coordinate which determines the horizontal position of the point and a y coordinate which determines the vertical position of the point.

The x-axis is measured from left to right (i.e. left is negative and right is positive) and the y-axis is measured from bottom to top.

For example the point ( 4, -2 ) is shown in the diagram below.

Equations of Lines

A line can be described by a linear equation in x and y the solutions of which form the points on a line. The standard form of the equation of a line is:

Where is the gradient (or slope) of the line and is the y intercept i.e. where the line crosses the y-axis.

For example the line with the equation is plotted on the graph below.

You can see that the y intercept is -3 and that the gradient is 2 (i.e. the line rises by 2 each time you move 1 space right).

32

1-4 -3 -2 -1

-1

-2

-3

1 2 3 4

Y

X

.

.

45

6

5 6

-5-6

-4-5-6

P

x-3

x=9.6

In the above figure, the equation y = 1 × x + 2, y = x + 2.

The difference in the y coordinates

The difference in the x coordinates

= 0−3

2−(−1−1

)=−3

3=−1

x intercept in then coordinate of the point at which the line intersects the x- axis.

Slope =0−5

6−0 =−5

6 −5

6

y=−56

×x+5

−3=−5 x6

+5

5x6

=8

5 x=48

x=485

= 9.6

∴ x=9. 6

(Short note)

In the xy plane the slope of the two parallel lines is same.

The expression

x3-5x2+2

f ( x )=x3−5 x2+2

f ( x )=1−5+2 ,f (1)=−2

f ( x )=|x+4|

then f (−5 )and f (−3 ) output same for equation.

y = Mx+6

9.6

f ( x )=−56x+6 [−5

6=slope ]

If the value of the function f ( x ) ,is equated with the variable y, then the graph of the

function in the xy –coordinate plane is simply the graph of the equation

y=−56

x+6

shown above. Similarly any function f ( x )

can be graphed by equating y with value of

function :

y=f ( x )

So far any x is the domain of functions f, the point with coordinates {x, f (x)} is on the

graph of f , and the graph consists entirely of these points.

Polynomial function f(x)-x2-1

X f (x)

-2 3

-1 0

0 -1

1 0

2 3

If all the points were graphed for−2≤x≤2 , then the graph would appear as follows

-1,0. .0,-1

1,0

-2,3. .2,3

-3

3

-2

21

+2 +3

The quadric function in the above graph is a parabola.

1. Chandler is building a fence in the following method: He grounds 10 poles, each

10 Cm thick, in 1 meter spaces from each other. He then connects the poles with a

barbed wire. What is the total length of the fence?

(a) 11.

(b) 12.

(c) 9.9.

(d) 10.

(e) 13.

The best answer is D.

2. Alfa, Beta and Gamma are inner angles in a triangle. If Alfa = Beta + Gamma,

what can’t be the size of Beta?

(a) 44 degrees.

(b) 45 degrees.

(c) 89 degrees.

(d) 90 degrees.

(e) There isn’t enough data to determine.


3. In a triangle, one side is 6 Cm and another side is 9 Cm. which of the following

can be the perimeter of the triangle?

(a) 18.

(b) 25.

(c) 30.

(d) 32.

(e) 34.

The best answer is B.

4. To which of the following shapes the area can’t be calculated if the perimeter is

given?

(a) Circle.

(b) An isoceles right triangle.

(c) Rectangle.

(d) A regular Hexagon.

(e) Square.

The best answer is C.

4. A and B are two circles. The radius of A is twice as large as the diameter of B.

What is the ratio between the areas of the circles?

(a) 1:8. (b) 1:2.

(c) 1:4.

(d) 1:16.

(e) 1:6.


5. A, B, C, D and E are 5 consecutive points on a straight line. If BC = 2CD, DE = 4,

AB = 5 and AC = 11, what is the length of AE?

(a) 21.

(b) 26.

(c) 30.

(d) 18.

(e) 16.


6. In a rectangular axis system, what is the distance between the following points: A(3,2) and

B(7,5) ?

(a) 5.

(b) 7.

(c) 6.

(d) 4.

(e) 3.

The best answer is A.

7. In a rectangular axis system, what is approximate distance between the following

points: C(1,2.5) and D(6.5,5.5) ?

(a) 5.5.

(b) 7.2.

(c) 6.3.

(d) 4.1.

(e) 3.8.


8. In a rectangular axis system, what is the distance between the following

points: A(24.4,30) and B(34.4,42.49) ?

(a) 5.

(b) 7.

(c) 8.

(d) 12.

(e) 16.


9. In a rectangular axis system, what is the area of a parallelogram with the

coordinates: (5,7), (12,7), (2,3), (9,3) ?

(a) 21.

(b) 28.

(c) 35.

(d) 49.

(e) 52.


10. If the radius of a cylinder is doubled and so is the height, what is the new volume

of the cylinder divided by the old one?

(a) 8.

(b) 2.

(c) 6.

(d) 4. (e) 10.


11. If the radius of a cylinder is doubled and so is the height, how much bigger is the

new lateral surface area (with out the bases)?

(a) 8.

(b) 2.

(c) 6.

(d) 4.

(e) 10.


12. (x, y) are the coordinates of the intersection of the following lines: (3x – 2y = 8) and (3y + x

= 10). What is the value of (x/y)?

A. 1

B. 2

C. 3

D. 4

E. 5


13. A (a, b) is the coordinates of the intersection between the lines: (x + y –1 = 0) and (4x – 2y =

5). What is the shortest distance between A(a, b) and the coordinate B(25/6, 23/6)?

A. 1

B. 2

C. 3

D. 4

E. 5

The best answer is E.

14. P(x, y) is the intersection point between the circle (x2 + y2 = 4) and the line (y = x +2).

Which of the following can be the point P?

A. (1, 2)

B. (2, 0)

C. (0, -2)

D. (-2, 0)

E. (2, 2)


15. Is the intersection of the two lines: (x + y = 8) and (4y – 4x = 16) inside the circle: x + y2 =

r2?

(1) r = 81.

(2) The center of the circle is at the coordinate (-99, -99)


16. Is there an intersection between the line (Y = aX - b) and the parabola (Y = X2 + b)?

(1) a < 0.

(2) 0 > b.


17. Is there a point of intersection between the circle (X2 + Y2 = 4) and the Line ( Y = aX + b) ?

(1) a = b2

.(2) The line intersects the X-axis at (40, 0).

18. What is the area of the rectangle with the following coordinates: (x, y), (10, y), (10, 5), (x, 5)? A .6. B. 8. C. 12. D. 32. E. It cannot be determined from the information given. The best answer is E.

19. What is the area of the square with the following coordinates: (x, y), (20, 20), (20, 5), (x, 5)? A. 60. B. 85. C. 125. D. 225. E. It cannot be determined from the information given. The best answer is D

20. The roof of an apartment building is rectangular and its length is 4 times longer than its width. If the area of the roof is 784 feet squared, what is the difference between the length and the width of the roof? A. 38. B. 40. C. 42. D. 44. E. 46. The best answer is C.

21. The length of a cube is three times its width and half of its height. If the volume of the Cube is 13,122 Cm cubed. What is the height of the cube? A. 49. B. 50. C. 54. D. 68. E. 81. The best answer is C.

22. The width of a cube is half the length and one third of the height. If the length of the cube is 4 meters, what is the volume of three identical cubes? A. 96. B. 88. C. 74. D. 68. E. 62. 23. Two adjacent angles of a parallelogram are in the ratio of 1:3. What is the smaller angle of the two? (a) 30. (b) 45. (c) 90. (d) 135. (e) 180. The best answer is B.

24. Two adjacent angles of a parallelogram are in the ratio of 2:3. What is their average size? (a) 30. (b) 40. (c) 45. (d) 90. (e) 180. The best answer is D.

25. The angles of a triangle are in the ratio of 3: 2: 1. The largest angle in the triangle is: (a) 36. (b) 45. (c) 72. (d) 90. (e). 108.

The best answer is E.

26. The perimeter of a circle is approximately 6.3 centimeters. The area of the same circle is A. which of the following is true? (a) 1 < A < 2 (b) 2 < A < 3. (c) 3 < A < 4. (d) 4 < A < 5. (e) A > 5.


27. One cubic centimeter is equal to 0.001 liters, is a volume of a rectangular tank larger than 0.001 liters? (1) The rectangular tank holds 0.3 teaspoons. There are 0.0049 liters in one teaspoon. (2) The dimensions of the tank are 0.5 x 0.6 x 4 centimeters. The best answer is D.

28. Two giant identical poles have been planted in the ground. One of the poles was planted dipper than the other pole. The shadow of pole A is 10 meters long and the shadow of pole B is 8 meters long. How tall is pole B? (1) Pole A is hoisted 14 meters in the air. (2) Pole B is located 2 meters from pole A The best answer is A.

29. Jean and Jordy each had to wash half of a rectangular floor. If Jean finished his part of the job after 45 minute, how long will it take Jordy to finish his half? (1) Jean can wash 10 meters square in 5 minutes, which is twice as fast as Jordy. (2) The area of the rectangular floor is 180 meters squared.


30. In a rectangular coordinate system, what is the area of a triangle whose vertices have the coordinates (4, 0), (6, 3), and (6, -3)?

(a) 7.5 (b) 7 (c) 6.5 (d) 6 (e) 5.5 The best answer is D.

31. In a rectangular coordinate system, what is the area of a rectangle whose vertices have the coordinates (-4, 1), (1, 1), (1, -3) and (-4, -3)? (a) 16 (b) 20 (c) 24 (d) 25 (e) 30 The best answer is B.

32. In a rectangular coordinate system, what is the area of a rhombus whose vertices have the coordinates (0, 3.5), (8, 0), (0, -3.5), (-8, 0)? (a) 56 (b) 88 (c) 112 (d) 116 (e) 120 The best answer is A. 33. In a rectangular coordinate system, what is the square root of the area of a trapezoid whose vertices have the coordinates (2, -2), (2, 3), (20, 2), (20, -2)? (a) 7.5 (b) 9 (c) 10.22 (d) 12.25

(e) 14 The best answer is B.

34.The line Y = 3X is drawn on a rectangular axis system. If the line is rotated, on which quadrant will it be found? (1) The rotation is done clockwise. (2) The line is rotated 180 degrees. The best answer is B.

35. In an isosceles triangle the sum of the sides is 2 inches longer than the base. What is the ratio between the length of the side and the length of the base? A. 1.5 B. 1. C. 1.75. D. 2. E. Not enough information. The best answer is E.

36. The perimeter of a rectangle is 136, what is the area of the rectangle?

(1) The length is more than twice the width.

(2) The length and width are both prime numbers larger than 30. The best answer is B.

37. Is the triangle ABC isosceles? (1) Angle A is equal to the sum of angles B and C. (2) Side AB is different from CB. The best answer is E.

38.Is the triangle ABC equilateral? (1) Angle A is half the sum of angles B and C.

(3) AC is equal to AB.


39. What is the sum of two angles in a triangle? (1) One of the angles is equal to the sum of the other two. (2) One of the sides is equal to the other. The best answer is E.

40. A rectangle is 14 cm long and 10 cm wide. If the length is reduced by x cms and its width is increased also by x cms so as to make it a square then its area changes by :

A. 4 B. 144 C. 12 D. 2 E. None of the above.

Ans : A

41. A plot of land is in the shape of a trapezium whose dimensions are given in the figure below :

Hence the perimeter of the field is a. 50 m b. 64 m c. 72 m d. 84 m e. None of the above

Ans : c

42. Four concentric ( having the same center ) circles with radii, x, 2x, 3x and 4x are drawn to form two rings A and B as shown in the figure.

Ratio of the area of inner ring A to the area of outer ring B is a. 1 : 2 b. 1 : 4 c. 2 : 3 d. 3 : 7 e. None of the above

Ans : D

43. If the area of two circles are in the ratio 169 : 196 then the ratio of their radii is a. 10 : 11 b. 11 : 12 c. 12 : 13 d. 13 : 14 e. None of the above

Ans : D

44. the xy-coordinate plane, points A and B both lie on the circumference of a circle whose center is O, and the length of AB equals the circle's diameter. If the (x,y) coordinates of O are (2,1) and the (x,y) coordinates of B are (4,6), what are the (x,y) coordinates of A?

a. (3, 3/2) b. (1, 2/2) c. (0, -4) d. (2/2, 1) e. (-1, -2/2)

Ans : C

45. If a rectangle's length and width are both doubled, by what percent is the rectangle's area increased?

a. 50 b. 100 c. 200

d. 300 e. 400

Ans : D

46. A rectangular tank 10" by 8" by 4" is filled with water. If all of the water is to be transferred to cube-shaped tanks, each one 3 inches on a side, how many of these smaller tanks are needed?

a. 9 b. 12 c. 16 d. 21 e. 39

Ans : B

47. Point Q lies at the center of the square base (ABCD) of the pyramid pictured above. The pyramid's height (PQ) measures exactly one half the length of each edge of its base, and point E lies exactly halfway between C and D along one edge of the base. What is the ratio of the surface area of any of the pyramid's four triangular faces to the surface area of the shaded triangle?

a. 3 :√2 b. √5:1 c. 4√3:3 d. 2√2:1 e. 8:√5

Ans : D

48. Find the coordinates of the point which divides the line joining (5, -2) and (9, 6) internally in the ratio 1 : 3.

a. (6, 0) b. (6, 3) c. (0, 6) d. (3, 6)

Ans : A

49. Find the number of triangles in an octagon. a. 326 b. 120 c. 56 d. Cannot be determined

Ans : C

50. Find the equation of a line whose intercepts are twice of the line 3x - 2y - 12 = 0 a. 3x - 2y = 24 b. 2x - 3y = 12 c. 2x - 3y = 24 d. None of these

Ans : A

51. Find the area of the sector covered by the hour hand after it has moved through 3 hours and the length of the hour hand is 7cm.

a. 77 sq.cm b. 38.5 sq.cm c. 35 sq.cm d. 70 sq.cm

Ans : B

52. Find the area of the triangle whose vertices are (-6, -2), (-4, -6), (-2, 5). a. 36 b. 18 c. 15 d. 30

Ans : C

53. A stairway 10ft high is such that each step accounts for half a foot upward and one-foot forward. What distance will an ant travel if it starts from ground level to reach the top of the stairway?

a. 30 ft b. 33 ft c. 10 ft d. 29 ft

Ans : D

54. Each interior angle of a regular polygon is 120 degrees greater than each exterior angle. How many sides are there in the polygon?

a. 6 b. 8 c. 12 d. 3

Ans : C

55. What is the area of the largest triangle that can be fitted into a rectangle of length 'l' units and width 'w' units?

a. lw/3 b. (2lw)/3 c. (3lw)/4 d. (lw)/2

Ans : D

56. Which of the following is inCorrect? a. An incentre is a point where the angle bisectors meet. b. The median of any side of a triangle bisects the side at right angle. c. The point at which the three altitudes of a triangle meet is the orthocentre d. The point at which the three perpendicular bisectors meet is the centre of the

circumcircle.

Ans : B

57. A and B are two points with the co-ordinates (-2, 0) and (0, 5). What is the length of the diagonal AC if AB form one of the sides of the square ABCD?

a. units b. units c. units d. units

Ans : B

58. What is the measure of the circum radius of a triangle whose sides are 9, 40 and 41? a. 6 b. 4 c. 24.5 d. 20.5

Ans : D

59. If the sum of the interior angles of a regular polygon measures up to 1440 degrees, how many sides does the polygon have?

a. 10 sides b. 8 sides c. 12 sides d. 9 sides

Ans : A

60. If ABC is a right angle triangle with angle A = 900 and 2s = a + b + c, where a > b > c where notations have their usual meanings, then which one of the following is Correct?

a. (s - b) (s - c) > s (s - a) b. (s - a) (s - c) > s (s - b)

c. (s - a) (s - b) < s (s - c) d. 4s (s - a) (s - b) (s - c) = bc

Ans : C

61. What is the measure of in radius of the triangle whose sides are 24, 7 and 25? a. 12.5 b. 3 c. 6 d. None of these

Ans : B

62. What is the circum radius of a triangle whose sides are 7, 24 and 25 respectively? a. 18 b. 12.5 c. 12 d. 14

Ans : B

63. A regular hexagon is inscribed in a circle of radius r cms. What is the perimeter of the regular hexagon?

a. 3r b. 6r c. r d. 9r

Ans : B

64. A 4 cm cube is cut into 1 cm cubes. What is the percentage increase in the surface area after such cutting?

a. 4% b. 300% c. 75% d. 400%

Ans : B

65. If the diagonal and the area of a rectangle are 25 m and 168 m2, what is the length of the rectangle?

a. 17 m b. 31 m c. 12 m d. 24 m

Ans : D

66. The surface area of the three coterminous faces of a cuboid are 6, 15, 10 sq.cm respectively. Find the volume of the cuboid.

a. 30 b. 20 c. 40 d. 35

Ans : A

67. If each interior angle of a regular polygon is 150 degrees, then it is a. Octagon b. Decagon c. Dodecagon d. Tetrahedron

Ans : C

68. A 5 cm cube is cut into as many 1 cm cubes as possible. What is the ratio of the surface area of the larger cube to that of the sum of the surface areas of the smaller cubes?

a. 1 : 6 b. 1 : 5 c. 1 : 25 d. 1 : 125

Ans : B

69. If the sides of a triangle measure 72, 75 and 21, what is the measure of its in radius? a. 37.5 b. 24 c. 9 d. 15

Ans : C

70. The circumference of the front wheel of a cart is 30 ft long and that of the back wheel is 36 ft long. What is the distance travelled by the cart, when the front wheel has done five more revolutions than the rear wheel?

a. 20 ft b. 25 ft c. 750 ft d. 900 ft

Ans : D

71. The area of a square field is 24200 sq m. How long will a lady take to cross the field diagonally at the rate of 6.6 km/hr?

a. 3 minutes b. 2 minutes c. 2.4 minutes d. 2 minutes 40 seconds

Ans : B

72. a and b are the lengths of the base and height of a right angled triangle whose hypotenuse is h. If the values of a and b are positive integers, which of the following cannot be a value of the square of the hypotenuse?

a. 13 b. 23 c. 37 d. 41

Ans : B

73. The angle of elevation of the top of a tower 30 m high, from two points on the level ground on its opposite sides are 45 degrees and 60 degrees. What is the distance between the two points?

a. 30 b. 51.96 c. 47.32 d. 81.96

Ans : C

74. A semi-circle is surmounted on the side of a square. The ratio of the area of the semi-circle to the area of the square is

a. 1 : 2 b. 2 : p c. p : 8 d. 8 : p e. None of the above

Ans : C

A

. B C 75. In the figure above, does x = 90? (1) The length of AC is less than the length of BC. (2) The length of AB is one-fourth the circumference of the circle.

78. The length of rectangle ABCD is 6/5th of its breadth. Its perimeter is 132. Find its area. A. 660 m² B. 2210 m² C. 1080 m² D. 2160 m² E. 460 m² Correct Answer: C

79. Given that AB and CD are two chords of a circle intersecting at the point O outside the circle, and AB = 7 m, BO = 5 m, and OD = 3 m. Find the length of OC. A. 20 m B. 16 m C. 35 m D. 6 m E. 18 m Correct Answer: A

B C

A D

x

80. In parallelogram ABCD above, what is the measure of ∠ADC?

(1) The measure of ∠ABC is greater than 90o. (2) The measure of ∠BCD is 70o 81. Is ΔMNP isosceles? (1) Exactly two of the angles, ∠M and ∠N, have the same measure (2) ∠N and ∠P do not have the same measure.

82. In ΔJKL shown above, what is the length of segment JL? (1) JK = 10 (2) KL = 5

83.The figure above represents the floor of a square foyer with a circular rug partially covering the floor and extending to the outer edges of the floor as shown. What is the area of the foyer that is not covered by the rug? (1) The area of the foyer is 9 square meters. (2) The area of the rug is 2.25π square meters.

84. Is quadrilateral Q a square? (1) The sides of Q have the same length. (2) The diagonals of Q have the same length.

85. The surface area of a square tabletop was changed so that one of the dimensions was reduced by 1 inch and the other dimension was increased by 2 inches. What was the surface area before these changes were made?

(1) After the changes were made, the surface area was 70 square inches. (2) There was a 25 percent increase in one of the dimensions.

86. A rectangular floor that is 4 meters wide is to be completely covered with non overlapping square tiles, each with side of length 0.25 meter, with no portion of any tile remaining. What is the least number of such tiles that will be required?

(1) The length of the floor is three times the width. (2) The area of the floor is 48 square meters.

86. What fractional parts of the total surface area of cube C is red?

(1) Each of 3 faces of C is exactly half red. (2) Each of 3 faces of C is entirely white.

87.In the figure above, D is a point on side AC of ΔABC. Is ΔABC is isosceles?

(1) The area of triangular region ABD is equal to the area of triangular region DBC. (2) BD┴AC and AD = DC

88. If the area of triangular region RST is 25, what is the perimeter of RST? (1) The length of one side of RST is 2 5 . (2) RST is a right isosceles triangle.

89.In ΔHGM, what is the length of side HM? (1) HG = 5 (2) GM = 8

90. Rectangle ABCD is inscribed in a circle as shown above. What is the radius of the circle?

(1) The length of the rectangle is 3 and the width of the rectangle is 1. (2) The length of are AB is one –third of the circumference of the circle.

91.

In the figure above, line AC represents a seesaw that is touching level ground at point A. If B is the midpoint of AC, how far above the ground is point C? (1) x = 30 (2) Point B is 5 feet above the ground.

92. Is ΔRST a right triangle? (1) The degree measure of ∠R is twice the degree measure of ∠T. (2) The degree measure of ∠T is 30.

93. The figure above shows four pieces of tile that have been glued together to form a square tile ABCD. Is PR=QS?

(1) BQ = CR = DS = AP (2) The perimeter of ABCD is 16.

94. What is the volume of a certain cube? (1) The sum of the areas of the faces of the cube is 54. (2) The greatest possible distance between two points on the cube is 3√3

95. A circular tub has a band painted around its circumference, as shown above. What is the surface area of this painted band? (1) x = 0.5 (2) The height of the tub is 1 meter.

96. What is the value of z in the triangle above? (1) x + y = 139 (2) y + z = 108

97. What is the circumference of the circle above? (1) The length of arc XYZ is 18. (2) r = s

98. In ΔPQR, if PQ = x, QR = x + 2, and PR = y, which of the three angles of ΔPQR has the greatest degree measure?

(1) y = x + 3 (2) x = 2

99. If l and w represent the length and width, respectively, of the rectangle above, what is the perimeter? (1) 2l + w = 40 (2) l + w = 25

100.

What is the radius of the circle above with center O? (1) The ratio of OP to PQ is 1 to 2. (2) P is the midpoint of chord AB.

101.

In the triangle above, does a2 + b2 = c2? (1) x + y = 90 (2) x = y

102. If ab ≠ 0, in what quadrant of the coordinate system above does point (a, b) lie?

(1) (b, a) lies in quadrant Ⅳ. (2) (a, -b) lies in quadrant Ⅲ.

103. In the figure above, segments PR and QR are each parallel to one of the rectangular coordinate axes. Is the ratio of the length of QR to the length of PR equal to 1?

(1) c = 3 and d = 4. (2) a = -2 and b = -1.

104. In the figure above, if lines k and m are parallel, what is the value of x? (1) y = 120 (2) z = 60

105. If P and Q are each circular regions, what is the radius of the larger of these regions? (1) The area of P plus the area of Q is equal to 90π. (2) The larger circular region has a radius that is 3 times the radius of the smaller circular region.

106. In the figure above, segments RS and TU represent two positions of the same ladder leaning against the side SV of a wall. The length of TV is how much greater than the length of RV? (1) The length of TU is 10 meters. (2) The length of RV is 5 meters.

107. In the figure above, what is the measure of ∠ABC ? (1) BX bisects ∠ABY and BY bisects ∠XBC. (2) The measure of ∠ABX is 40o

108. The inside of a rectangular carton is 48 centimeters long, 32 centimeters wide, and 15 centimeters high. The carton is filled to capacity with k identical cylindrical cans of fruit that stand upright in rows and columns, as indicated in the figure above. If the cans are 15 centimeters high, what is the value of k? (1) Each of the cans has a radius of 4 centimeters. (2) 6 of the cans fit exactly along the length of the carton.

109. What is the area of the rectangular region above? (1) l + w = 6 (2) d2 = 20

110. If point X is inside a circle with center O and radius 2, is point Y inside the same circle? (1) OX = 1

(2) XY = 212

111. In the figure above, what is the length of segment BC ? (1) x = 90 (2)The perimeter of △ABC is 24.

112. If y = ax + b, where a and b are constants, what is the value of y when x = 10 ? (1) When x = 1, y = 5. (2) When x = 5, y = 13.

113. In the figure above, x, y, and z denote the lengths of the sides of a triangular flower bed bounded by three driveways. What is the perimeter of the flower bed ? (1) x = y = 30 feet.

(2) k = 60

114. In the rectangular coordinate system, if line l is parallel to one of the axes, does line l contain the point (4, 5) ? (1) Line l contains the point (4, -5). (2) Line l crosses the x-axis.

114. In the figure above, if PR is a line segment, what is the sum of the lengths of the curved paths from P to Q and from Q to R? (1) XQ = QY = 5 centimeters. (2) Every point on arc PQ is 5 centimeters from point X, and every point on arc QR is 5 centimeters from point Y.

115. In the figure above, what is the area of the circular region with center O ? (1) MN is perpendicular to RS. (2) The area of triangular region OMT is 4.

116. If P, Q, and R are three distinct points, do line segments PQ and PR have the same length? (1) P is the midpoint of line segment QR. (2) Q and R lie on the same circle with center P.

117. The length of the edging that surrounds circular garden K is 21 the length of the edging that surrounds circular garden G. What is the area of garden K ? (Assume that the edging has negligible width.) (1) The area of G is 25π square meters. (2) The edging around G is 10π meters long.

118. Quadrilateral RSTU shown above is a site plan for a parking lot in which side RU is parallel to side ST and RU is longer than ST. What is the area of the parking lot ? (1) RU = 80 meters (2) TU= 10 20 meters

119. The figure above represents the floor plan of an art gallery that has a lobby and 18 rooms. If Lisa goes from the lobby into room A at the same time that Paul goes from the lobby into room R, and each goes through all of the rooms in succession, entering by one door and exiting by the other, which room will they be in at the same time ? (1) Lisa spends 2x minutes in each room and Paul spends 3x minutes in each room. (2) Lisa spends 10 minutes less time in each room than Paul does.

NUMBER THEORY

If x and y are nonzero integers, is xy

an integer?

(1) x is the product of 2 and some other integer. (2) There is only one pair of positive integers whose product equals y.

If n is an integer between 2 and 100 and if n is also the square of an integer, what is the value of n? (1) n is the cube of an integer. (2) n is even.

. Is x2 – y2 a positive number? (1) x – y is a positive number. (2) x + y is a positive number.

. If x is an integer, what is the value of x?

(1) 15< 1x+1

< 12

(2)( x−3 )( x−4)=0

If K is a positive integer less than 10 and N = 4,321 + K, what is the value of K? (1) N is divisible by 3. (2) N is divisible by 7.

Is x – y > r – s? (1) x > r and y < s? (2) y = 2, s = 3, r = 5, and x = 6.

What is the value of x? (1) x + y = 7 (2) x – y = 3 – y

Is n an integer greater than 4? (1) 3n is a positive integer.

(2) n3

is a positive integer.

Is x2equal to xy? (1) x2 – y2 = (x + 5)(y - 5)

(2) x = y

What is the value of x –1? (1) x + 1 =3 (2) x – 1 < 3

What fraction of his salary did Mr. Johnson put into savings last week? (1) Last week Mr. Johnson put $17 into savings. (2) Last week Mr. Johnson put 5% of his salary into savings.

If a rope is cut into three pieces of unequal length, what is the length of the shortest of these pieces of rope? (1) The combined length of the longer two pieces of rope is 12 meters. (2) The combined length of the shorter two pieces of rope is 11 meters.

PERCENTAGEAt a certain university, if 50 percent of the people who inquire about admission policies actually submit applications for admission, what percent of those who submit applications for admission enroll in classes at the university? (1) Fifteen percent of those who submit applications for admission are accepted at the university. (2) Eighty percent of those who are accepted send a deposit to the university.

1. If today the price of an item is $3,600, what was the price of the item exactly 2 years ago?

(1) The price of the item increased by 10 per-cent per year during this 2-year period. (2) Today the price of the item is 1.21 times its price exactly 2 years ago.

2. By what percent has the price of an overcoat been reduced? (1) The original price was $380. (2) The original price was $50 more than the reduced price.

A jewelry dealer initially offered a bracelet for sale at an asking price that would give a profit to the dealer of 40 percent of the original cost. What was the original cost of the bracelet? (1)After reducing this asking price by 10 percent, the jewelry dealer sold the bracelet at a profit of $403.

(2) The jewelry dealer sold the bracelet for $1,953.

AVARAGE

A coal company can choose to transport coal to one of its customers by railroad or by truck. If the railroad charges by the mile and the trucking company charges by the ton, which means of transporting the coal would cost less than the other? (1) The railroad charges $5,000 plus $0.01 per mile per railroad car used, and the trucking company charges $3,000 plus $85 per ton. (2) The customer to whom the coal is to be sent is 195 miles away from the coal company.

Was 70 the average (arithmetic mean) grade on a class test? (1) On the test, half of the class had grades below 70 and half of the class had grades above 70. (2) The lowest grade on the test was 45 and the highest grade on the test was 95.

. What was John’s average driving speed in miles per hour during a 15-minute interval? (1) He drove 10 miles during this interval. (2) His maximum speed was 50 miles per hour and his minimum speed was 35 miles per hour during this interval.

What is the average distance that automobile D travels on one full tank of gasoline? (1) Automobile D averages 8.5 kilometers per liter of gasoline. (2) The gasoline tank of automobile D holds exactly 40 liters of gasoline.A certain company paid bonuses of $125 to each of its executive employees and $75 to each of its nonexecutive employees. If 100 of the employees were nonexecutives, how many were executives? (1) The company has a total of 120 employees. (2) The total amount that the company paid in bonuses to its employees was $10.000.

TIME AND DISTANCE

On a certain day it took Bill three times as long to drive from home to work as it took Sue to drive from home to work. How many kilometers did Bill drive from home to work?

(1) Sue drove 10 kilometers from home to work, and the ratio of distance driven from home to work time to drive from home to work was the same for Bill and Sue that day.

(2) The ratio of distance driven from home to work time to drive from home to work for Sue that day was 64 kilometers per hour.

TIME AND WORKIn a certain packinghouse, grapefruit are packed in bags and the bags are packed in cases. How many grapefruit are in each case that is packed? (1) The grapefruit are always packed 5 to a bag and the bags are always packed 8 to a case. (2) Each case is always 80 percent full.

Who types at a faster rate, John or Bob? (1) The difference between their typing rates is 10 words per minute.

(2) Bob types at a constant rate of 80 words per minute.

Is William taller than Jane? (1) William is taller than Anna. (2) Anna is not as tall as Jane.

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This Tutorial Will Cover Everything You Need to Know About Geometry for the GMAT