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GEOMETRY • MP2 Q2 TEST FEB 5th • LAST TEST OF MP2 • REVIEW QUESTIONS/TOPICS: The MP2 Quarterly Assessment will be graded as the last test for the second marking period. This district exam will be given purely online and will not be eligible for a retake. USE EXTRA PAPER TO SOLVE. 1. Similarity diagram and word problem-‐ explain why similar and find missing value a. A swimmer needs to know the width of a river without having to cross it. She made a diagram below.
Note: This figure is not drawn to scale. What is the width (w), in meters, of the river?
b. Campsites F and G are on opposite sides of a lake. A survey crew made the measurements shown on the
diagram. What is the distance between the two campsites? The diagram is not to scale.
2. The following pairs of triangles are similar. Find the missing values. a. b. c.
3. Find many missing angles in involved diagram
Fill in each proof: 11. Given: B E#( ( , and are right anglesA D( (
Prove: BC ABEC DE
12. Given: JK GH&
Prove: FJ FKFG FH
13. A lighthouse casts a 128-foot shadow. A nearby lamppost that measures 5 feet 3 inches casts an 8-foot shadow. A. 5 feet 3 inches is 5.___________ feet.
B. Write a proportion that can be used to determine the height of the lighthouse. C. What is the height of the lighthouse?
Find the little to big ratio, set up a proportion, and solve for each variable. 14. Little to big ratio:
Proportion to solve for x: x = ____________
15. Little to big ratio:
Proportion to solve for x: Proportion to solve for y: x = _____________ y = ______________
16. Little to big ratio:
Proportion to solve for x: Proportion to solve for y: x = _____________ y = ______________
17. Little to big ratio:
Proportion to solve for x: x = ____________
18. Little to big ratio:
Proportion to solve for x: x = ____________ ED = ___________
19. Little to big ratio:
Proportion to solve for x: x = ____________ BC = ___________ AB = ___________
20. Little to big ratio:
Proportion to solve for x: x = ____________ AB = ___________
21. Little to big ratio:
Proportion to solve for x: x = ____________ FH = ___________
22. Little to big ratio:
Proportion to solve for x: x = ____________ MN = ___________
23. Little to big ratio:
Proportion to solve for x: x = ____________ AC = ___________
Marking(Period(2(Assessment(Review((Page(4)((Unit%5%'%Similarity%(Con’t)%(9a)(( ( Dilate(the(triangle(with(center(of(dilation(at(the(( ( ( ( origin(and(a(scale(factor(of(1/3.(((((((((((((((9b)(( ( Dilate(the(figure(with(center(of(dilation((2,(2)(( ( ( ( and(a(scale(factor(of(½.((((((((((((((10)(The(triangles(are(similar.((Set(up(proportions(to(solve(for(x(and(y.(((( ( 9( 12( ( 5(((( y( 4(((( x((((((
Marking(Period(2(Assessment(Review((Page(3)((Unit%3%'%Triangles%(Con’t)%(7)(Find(the(requested(angles:((( a)(( m<U(=( b)(( m<U(=((( ( ( m<VUT(=( ( m<VUT(=(((((((((( c)( ?(=( d)( ?(=(((((((Unit%5%'%Similarity%(8a)(( ( What(is(the(scale(factor?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( angles?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( sides?(((((8b)(( ( What(is(the(scale(factor?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( angles?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( sides?(((
4. Transformations (especially 90˚ Rotation) a) perform the following transformation: b) rotate 90 degrees CCW
(x, y) → (x + 2, y −3)
c) Find the coordinates of the image of a 90˚ rotation of pre-‐image: A(-‐4, 2), B(3, -‐7), and C(-‐2, -‐6) 5. Perform dilations/find the scale factor in coordinate plane Dilate each figure by the given scale factor with P as the center of dilation.
6. Slope of a line perpendicular to two given points
a.
b. 7. Find the coordinate that will divide the segment to a 1:1 ratio a. b.
©Y T2B0U1R0P TKfuntNas VSZoKfNtzw8akrBeO BLUL2Cr.l w TAFlSly XreiFg4hAtIsC brTeDsce4rXvYeldb.G W 5MfaIdYeZ jwaiDt6hX bICn8fmi6nUiNtMer XGQeNoKmleetlr8yD.x Worksheet by Kuta Software LLC
Kuta Software - Infinite Geometry Name___________________________________
Period____Date________________All Transformations
Graph the image of the figure using the transformation given.
1) rotation 90° counterclockwise about theorigin
x
y
J
Z
L
2) translation: 4 units right and 1 unit down
x
y
Y
F
G
3) translation: 1 unit right and 1 unit up
x
y
E
J
T
M
4) reflection across the x-axis
x
y
M
C J
K
Write a rule to describe each transformation.
5)
x
y
H
C
B
H'
C'
B'
6)
x
y
P
D
E
I
D'
E'
I' P'
-1-
Marking(Period(2(Assessment(Review((Page(2)((Unit%2%'%Angles%and%Lines%(Con’t)%(4)(Lines(m(and(n(are(cut(by(transversal(p.((What(type(of(angle(relationships(are(the(following(pairs((alternate(interior,(same;side(interior,(alternate(exterior,(corresponding,(vertical)?((a)( 1(and(2:((( 2(and(3:((( 3(and(4:((( 1(and(4:( (((b)( 1(and(2:((( 2(and(3:((( 3(and(4:((( 1(and(4:( (((5a)( What(is(the(slope(of(a(line(that(is(perpendicular(to(the(line(that(passes(through(( the(points((3,(7)(and((8,(3)?((((5b)( What(is(the(slope(of(a(line(that(is(perpendicular(to(the(line(that(passes(through(( the(points((4,(8)(and((9,(4)?((((Unit%3%'%Triangles%(6)(Answer(the(following(questions(regarding(right(triangles:((( a)(How(many(acute(angles?((( b)(How(many(right(angles?((( c)(How(many(obtuse(angles?((( d)(How(are(the(acute(angles(related?((( e)(What(is(the(relationship(between(the(legs(and(the(hypotenuse?(((
Marking(Period(2(Assessment(Review((Page(2)((Unit%2%'%Angles%and%Lines%(Con’t)%(4)(Lines(m(and(n(are(cut(by(transversal(p.((What(type(of(angle(relationships(are(the(following(pairs((alternate(interior,(same;side(interior,(alternate(exterior,(corresponding,(vertical)?((a)( 1(and(2:((( 2(and(3:((( 3(and(4:((( 1(and(4:( (((b)( 1(and(2:((( 2(and(3:((( 3(and(4:((( 1(and(4:( (((5a)( What(is(the(slope(of(a(line(that(is(perpendicular(to(the(line(that(passes(through(( the(points((3,(7)(and((8,(3)?((((5b)( What(is(the(slope(of(a(line(that(is(perpendicular(to(the(line(that(passes(through(( the(points((4,(8)(and((9,(4)?((((Unit%3%'%Triangles%(6)(Answer(the(following(questions(regarding(right(triangles:((( a)(How(many(acute(angles?((( b)(How(many(right(angles?((( c)(How(many(obtuse(angles?((( d)(How(are(the(acute(angles(related?((( e)(What(is(the(relationship(between(the(legs(and(the(hypotenuse?(((
©U X2P0l1y1z EKZuLtcae cSjoifXtKwOaWrIe7 4LlLiCB.d 8 aA7l1ld 8rbitgphAtysP 2r4eEspenr9vJeRdQ.A M jMualdBeD TwCiWtshC eIxn8ftiOnmipt0e8 sPnr0eW-HAclRgBeWbmrka7.W Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________Rotations of Shapes
Graph the image of the figure using the transformation given.
1) rotation 180° about the origin
x
y
J
Q
H
2) rotation 90° counterclockwise about theorigin
x
y
S
B
L
3) rotation 90° clockwise about the origin
x
y
M
B
F
H
4) rotation 180° about the origin
x
y
U
H
F
5) rotation 90° clockwise about the origin
U(1, −2), W(0, 2), K(3, 2), G(3, −3)
x
y
6) rotation 180° about the origin
V(2, 0), S(1, 3), G(5, 0)
x
y
-1- ©U X2P0l1y1z EKZuLtcae cSjoifXtKwOaWrIe7 4LlLiCB.d 8 aA7l1ld 8rbitgphAtysP 2r4eEspenr9vJeRdQ.A M jMualdBeD TwCiWtshC eIxn8ftiOnmipt0e8 sPnr0eW-HAclRgBeWbmrka7.W Worksheet by Kuta Software LLC
Kuta Software - Infinite Pre-Algebra Name___________________________________
Period____Date________________Rotations of Shapes
Graph the image of the figure using the transformation given.
1) rotation 180° about the origin
x
y
J
Q
H
2) rotation 90° counterclockwise about theorigin
x
y
S
B
L
3) rotation 90° clockwise about the origin
x
y
M
B
F
H
4) rotation 180° about the origin
x
y
U
H
F
5) rotation 90° clockwise about the origin
U(1, −2), W(0, 2), K(3, 2), G(3, −3)
x
y
6) rotation 180° about the origin
V(2, 0), S(1, 3), G(5, 0)
x
y
-1-
Marking(Period(2(Assessment(Review((Page(1)((Unit%1%'%Transformations%(1a)( A(2,(0),(B(1,(;3),(C(6,(;2)( 1b)(( D(0,(3),(E(;1,(4),(F(;2,(;3)(( (x,(y)!(x(;(5,(y(+(3)( (x,(y)!(x(+(4,(y(;(2)(( Draw(the(pre;image(and(image.( Draw(the(pre;image(and(image.(( What(effect(did(the(transformation(have?( What(effect(did(the(transformation(have?((( ((((((((((2a)( ∆ABC(=(A(2,(1),(B(3,(5),(C(1,(3).( 2b)( ∆DEF(=(D(3,(0),(E(4,(4),(F(3,(2).(( Rotate(∆ABC(90˚(ccw(about(the(origin.( ( Rotate(∆DEF(90˚(ccw(about(the(origin.(( Then(reflect(it(across(the(x;axis.( ( Then(reflect(it(across(the(x;axis.(( Label(your(final(image(points(as(A’’B’’C’’.( ( Label(your(final(image(points(as(D’’E’’F’’.(( A”(=((((((,((((),(B”(=((((((,((((),(C”(=((((((,(((().( ( D”(=((((((,((((),(E”(=((((((,((((),(F”(=((((((,(((().((( ( (
((( ( ( (((((((Unit%2%'%Angles%and%Lines%(3a)( Find(the(point(M(along(the(directed(line( 3b)( Find(the(point(M(along(the(directed(line(( segment(that(divides(the(line(segment( ( segment(that(divides(the(line(segment(( in(the(ratio(1(to(1((it’s(the(midpoint).( ( in(the(ratio(1(to(1((it’s(the(midpoint).((( M((midpoint)(=((((((,(((()( M((midpoint)(=((((((,(((()((((( (((((
8. Two lines cut by a transversal (not parallel), ID types of angles
9. Dilations and scale factor in the coordinate plane
Marking(Period(2(Assessment(Review((Page(2)((Unit%2%'%Angles%and%Lines%(Con’t)%(4)(Lines(m(and(n(are(cut(by(transversal(p.((What(type(of(angle(relationships(are(the(following(pairs((alternate(interior,(same;side(interior,(alternate(exterior,(corresponding,(vertical)?((a)( 1(and(2:((( 2(and(3:((( 3(and(4:((( 1(and(4:( (((b)( 1(and(2:((( 2(and(3:((( 3(and(4:((( 1(and(4:( (((5a)( What(is(the(slope(of(a(line(that(is(perpendicular(to(the(line(that(passes(through(( the(points((3,(7)(and((8,(3)?((((5b)( What(is(the(slope(of(a(line(that(is(perpendicular(to(the(line(that(passes(through(( the(points((4,(8)(and((9,(4)?((((Unit%3%'%Triangles%(6)(Answer(the(following(questions(regarding(right(triangles:((( a)(How(many(acute(angles?((( b)(How(many(right(angles?((( c)(How(many(obtuse(angles?((( d)(How(are(the(acute(angles(related?((( e)(What(is(the(relationship(between(the(legs(and(the(hypotenuse?(((
Marking(Period(2(Assessment(Review((Page(3)((Unit%3%'%Triangles%(Con’t)%(7)(Find(the(requested(angles:((( a)(( m<U(=( b)(( m<U(=((( ( ( m<VUT(=( ( m<VUT(=(((((((((( c)( ?(=( d)( ?(=(((((((Unit%5%'%Similarity%(8a)(( ( What(is(the(scale(factor?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( angles?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( sides?(((((8b)(( ( What(is(the(scale(factor?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( angles?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( sides?(((
Marking(Period(2(Assessment(Review((Page(3)((Unit%3%'%Triangles%(Con’t)%(7)(Find(the(requested(angles:((( a)(( m<U(=( b)(( m<U(=((( ( ( m<VUT(=( ( m<VUT(=(((((((((( c)( ?(=( d)( ?(=(((((((Unit%5%'%Similarity%(8a)(( ( What(is(the(scale(factor?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( angles?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( sides?(((((8b)(( ( What(is(the(scale(factor?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( angles?(((( ( ( ( What(is(the(relationship(between(the(corresponding(( ( ( ( sides?(((
10. Solve the similar triangles
a. b.
c. d. 11. Right triangles
12. Congruent Triangles Explain why the following triangles are congruent. Give a justification for each step. a. Given: M is the midpoint of AB and what’s marked off in the diagram
b.
348 MHR • Chapter 7
For help with questions 5 to 7, see Example 1.
5. a) Show why !PQR is similar to !STR.
b) Find the lengths x and y.
6. The triangles in each pair are similar. Findthe unknown side lengths.
a)
b)
c)
d)
e)
7. Find the length of x in each.
a)
b)
For help with question 8, see Examples 2 and 3.
8. a) !PQR ~ !STU. Find the area of !PQR.
b) !ABC ~ !DEF. Find the area of !ABC.
c) !GHI ~ !KLM. Find the area of !KLM.
d) !STU ~ !XYZ. Find the area of !STU.
S
X
T U
8 cm10 cm
Y Z
A = 40 cm2
4 cm 6 cm
G
H I
L M
K
A = 12 cm2
ECB
8 cm12 cm
AD
F
A = 54 cm2
T
R
QP12 cm 9 cm
A = 72 cm2
S
U
x
A
B C
4 cm
6 cm
10 cm
D E
4 cm
6 cm
3 cm
5 cm
x
QP
R
S T
C
A
B
15 cm
12 cm18 cm
d
10 cme
D
E
D
FE10 cm
9 cm8 cm
R
Q
P
r
p
6 cm
5 cm
X
Y
W4 cm
w9 cm
C
A
B6 cm
b
T
R
S 8 cm
s
R
P
Q 24 cm
18 cm10 cm
r
A
B C
6 cm
7 cm
4 cm
D
E F
12 cm
d
f
15 cm27 cm
12 cm
18 cm
RP S
T
Q
xy
�David W. Sabo (2003) Solving Problems with Similar Triangles Page 1 of 6
B
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ADx
15
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The Mathematics 11 Competency Test Solving Problems with Similar
Triangles In the previous document in this series, we defined the concept of similar triangles, 'ABC a 'A’B’C’ as a pair of triangles whose sides and angles could be put into correspondence in such a way that it is true that property (i): A = A’ and B = B’ and C = C’.
property (ii): ' ' '
a b ca b c
If property (i) is true, property (ii) is guaranteed to be true. If property (ii) is true, then property (i) is guaranteed to be true. We also demonstrated some strategies for establishing that two triangles are similar using property (i). This is very useful to be able to do, since then, we may be able to use the property (ii) conditions to calculate unknown lengths in the triangles. Example 1: Given that lines DE and AB are parallel in the figure to the right, determine the value of x, the distance between points A and D. solution: First, we can demonstrate that 'CDE a 'CAB because C = C (by identity) and �CDE = �CAB because line AC acts as a transversal across the parallel lines AB and DE, and since �CDE and �CAB are corresponding angles in this case, they are equal. Since two pairs of corresponding angles are equal for the two triangles, we have demonstrated that they are similar triangles. To avoid error in exploiting the similarity of these triangles, it is useful to redraw them as separate triangles:
Marking(Period(2(Assessment(Review((Page(2)((Unit%2%'%Angles%and%Lines%(Con’t)%(4)(Lines(m(and(n(are(cut(by(transversal(p.((What(type(of(angle(relationships(are(the(following(pairs((alternate(interior,(same;side(interior,(alternate(exterior,(corresponding,(vertical)?((a)( 1(and(2:((( 2(and(3:((( 3(and(4:((( 1(and(4:( (((b)( 1(and(2:((( 2(and(3:((( 3(and(4:((( 1(and(4:( (((5a)( What(is(the(slope(of(a(line(that(is(perpendicular(to(the(line(that(passes(through(( the(points((3,(7)(and((8,(3)?((((5b)( What(is(the(slope(of(a(line(that(is(perpendicular(to(the(line(that(passes(through(( the(points((4,(8)(and((9,(4)?((((Unit%3%'%Triangles%(6)(Answer(the(following(questions(regarding(right(triangles:((( a)(How(many(acute(angles?((( b)(How(many(right(angles?((( c)(How(many(obtuse(angles?((( d)(How(are(the(acute(angles(related?((( e)(What(is(the(relationship(between(the(legs(and(the(hypotenuse?(((
218 Chapter 4 Congruent Triangles
PROOF Write a two-column proof or a paragraph proof.
25. GIVEN ! ACÆ £ BCÆ, 26. GIVEN ! BCÆ £ AEÆ, BDÆ £ ADÆ, M is the midpoint of ABÆ. DEÆ £ DCÆ
PROVE ! ¤ACM £ ¤BCM PROVE ! ¤ABC £ ¤BAE
27. GIVEN ! PAÆ £ PBÆ £ PCÆ, 28. GIVEN ! CRÆ £ CSÆ, QCÆ fi CRÆ,ABÆ £ BCÆ QCÆ fi CSÆ
PROVE ! ¤PAB £ ¤PBC PROVE ! ¤QCR £ ¤QCS
29. TECHNOLOGY Use geometry software to draw a triangle. Draw a lineand reflect the triangle across the line. Measure the sides and the angles
of the new triangle and tell whether it is congruent to the original one.
Writing Explain how triangles are used in the object shown to make itmore stable.
30. 31.
32. CONSTRUCTION Draw an isosceles triangle with vertices A, B, and C.Use a compass and straightedge to construct ¤DEF so that ¤DEF £ ¤ABC.
USING ALGEBRA Use the Distance Formula and the SSS CongruencePostulate to show that ¤ABC £ ¤DEF.
33. 34. 35.y
x1
2
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B
D
C
AE
y
x5
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A B
C DE
F
xyxy
SR
q
C
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A B
C
B
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A
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A BM
y
x1
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BA
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SOFTWARE HELPVisit our Web site
www.mcdougallittell.comto see instructions forseveral softwareapplications.
INTE
RNET
STUDENT HELP
6. Developing Proof Complete the proof.
Given: ∠1 ≅ ∠2, AB ⊥ BL , KL ⊥ BL , AB ≅ KL
Prove: ΔABG ≅ ΔKLG
Proof:
7. Write a flow proof.
Given: ∠E ≅ ∠H
∠HFG ≅ ∠EGF
Prove: ΔEGF ≅ ΔHFG
8. Write a two-column proof.
Given: ∠K ≅ ∠M
KL ≅ ML
Prove: ΔJKL ≅ ΔPML
For Exercises 9 and 10, write a paragraph proof.
9. Given: ∠D ≅ ∠G
HE ≅ FE
Prove: ΔEFG ≅ ΔEHD
10. Given: JM bisects ∠J.
JM ⊥ KL
Prove: ΔJMK ≅ ΔJML
Prentice Hall Gold Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
24
Name Class Date
Practice (continued) Form G 4-3 Triangle Congruence by ASA and AAS