prove: fg fh geometry(•mp2q2testfeb5 prove:...

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 = ______________


Proportion to solve for x: Proportion to solve for y: x = _____________ y = ______________


Proportion to solve for x: x = ____________


Proportion to solve for x: x = ____________ ED = ___________


Proportion to solve for x: x = ____________ BC = ___________ AB = ___________


Proportion to solve for x: x = ____________ AB = ___________


Proportion to solve for x: x = ____________ FH = ___________


Proportion to solve for x: x = ____________ MN = ___________


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.

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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'

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-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?(((


©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


1) rotation 180° about the origin

x

y

J

Q

H


x

y

S

B

L

3) rotation 90° clockwise about the origin

x

y

M

B

F

H


x

y

U

H

F


U(1, −2), W(0, 2), K(3, 2), G(3, −3)

x

y


V(2, 0), S(1, 3), G(5, 0)

x

y

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Kuta Software - Infinite Pre-Algebra Name___________________________________

Period____Date________________Rotations of Shapes



x

y

J

Q

H


x

y

S

B

L


x

y

M

B

F

H


x

y

U

H

F


U(1, −2), W(0, 2), K(3, 2), G(3, −3)

x

y


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




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

E

C

ADx

15

11

7

B

C

A

11

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7

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:


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

F

B

D

C

AE

y

x5

1

A B

C DE

F

xyxy

SR

q

C

P

A B

C

B

D

A

C

E

C

A BM

y

x1

1

F

C

D

BA

E

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

prove: fg fh geometry(•mp2q2testfeb5 prove:...

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