Topic ReviewTOPIC ESSENTIAL QUESTION7

TOPIC

Vocabulary Review

1. How are properties of similar figures used to solve problems?

Choose the correct term to complete each sentence.

2. Two triangles that are have two pairs of corresponding congruent angles.

3. A is a composition of a dilation and one or more rigid motions.

4. A point that is its own image in a dilation is the .

5. As a result of a dilation, if A′B′ = n ∙ AB , then n is the .

Quick ReviewA dilation is a transformation that maps a point X to X′ such that X′ lies on

⟶ CX and CX′ = k ∙ CX ,

with center of dilation C and scale factor k.

Two figures are similar if there is a similarity transformation that maps one figure onto  the other.

Practice & Problem SolvingGive the coordinates of each image.

6. D 1 _ 2 ( △FGH ) for F(5, −2 ), G( −2 , −4 ), H(0, 6)

7. D (3, K) ( △KLM ) for K(0, 4), L(3, 0), M( −2 , 4)

8. What is a similarity transformation from PQRS to WXYZ?

9. Construct Arguments Isabel says that the scale factor in the similarity transformation that maps △ABC to △PQR is 2. Is she correct? Explain.

7

5 14

10 8

4

P Q A

B

C

R

ExampleAre △ABC and △DEF similar? Explain.

4

2

–2 2 4

y

xO

B

A C

E

F D

The reflection R y-axis maps △ABC to a triangle with vertices A′(2, 0), B′(1, 2) and C′(1, 0) . The dilation D 2 maps the image to △DEF . Since the composition D 2 ∘ R y-axis maps △ABC to △DEF , the triangles are similar.

Concepts & Skills Review

LESSONS 7-1 & 7-2 Dilations and Similarity Transformations

• center of dilation

• dilation

• geometric mean

• scale factor

• similar

• similarity transformation

2

–2

–2 2

y

xO

W

Z Y

X

P

SR

Q

340 TOPIC 7 Similarity Go Online | PearsonRealize.com

TO

PIC

7 R

EV

IEW

Quick ReviewA pair of triangles can be shown to be similar by using the following criteria.• Two pairs of corresponding angles are

congruent.

• All corresponding sides are proportional.

• Two pairs of corresponding sides are proportional and the included angles are congruent.

Quick ReviewFor right triangle △ABC , △ABC ∼ △ACD ∼ △CBD .

Also, CD is the geometric mean of AD and BD, AC is the geometric mean of AB and AD, and CB is the geometric mean of AB and DB.

For △FGH, FJ ___ HJ

= FG ___ HG

.

Practice & Problem SolvingFor Exercises 10 and 11, explain whether each triangle similarity is true.

Practice & Problem SolvingFor Exercises 13–15, use △RST to find each length.

13. RS

14. ST

15. SU

For Exercises 16–19, find the value of x.

ExampleExplain whether △ABE and △DBCare similar.

By the Alternate Interior Angles Theorem, ∠A ≅ ∠D and ∠E ≅ ∠C . Since two pairs of corresponding angles are congruent, △ABE ∼ △DBC .

ExampleFor △LMN , what is x?

L N

QP

M9

6 4

x

By the Side-Splitter Theorem, LP ___ PM = NQ ___

QM ,

so 6 __ 9 = 4 __ x . Solve for x to get x = 9 __

6 (4) = 6 .

LESSON 7-3 Proving Triangles Similar

LESSONS 7-4 & 7-5 Similarity in Right Triangles and Proportions in Triangles

10. △FGJ ∼ △JGH

F J

HG

96

4

11. △KLN ∼ △NLM

L

K

N M

48

6

3

2

12. Communicate Precisely Explain what additional information is needed to use AA ∼ to show that △TUV ∼ △XZY .

T V

U

Z

X Y

37° 65°

78°

16. 2032

30x

17.

2018

16x

18.

2.5

58

x

19.

9

6

5x

20. Use Structure Given right triangle △ GHJ with ‾ JK the altitude to hypotenuse ‾ GH , what is GJ the geometric mean of? Explain.

A E

BC D

A

C B

D

F HJ

G

R TU

S

4 12

TOPIC 7 Topic Review 341