Dilations - IT'S TRIMBLE TIME€¦ · Name PearsonRealize.com 7-1 Reteach to Build Understanding Dilations 1. The diagram shows a dilation of ABC with a scale factor of 2 and the

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7-1 Reteach to Build UnderstandingDilations

1. The diagram shows a dilation of △ABC with a scale factor of 2 and the center of dilation at point X. Use the diagram to decide which statements are true. Label true statements T and false statements F.

m∠BAC = m∠B′A′C′ BC = 2(B′C′)

2(m∠ABC) = m∠A′B′C′ 2(XA) = XA′

2. Leah tried to graph D 1 __ 3 (DEFG).

Her graph is incorrect. List the correct coordinates for the vertices of the image.

D′( , ) E′( , )

F′( , ) G′( , )

3. Fill in the blanks about D (4, L) (△JKL), where J(−1, 5), K(4, 2), and L(0, −2).

The center of dilation is .

The scale factor of the dilation is .

Length J′K′ is JK.

The measure of ∠L is the measure of ∠L′.

Complete the table for D (4, L) (△JKL), where J(−1, 5), K(4, 2), L(0, −2).

J(−1, 5)

K(4, 2)

J′(−4, 26)

K′(16, 14)

−1

4

7

4

ImagePreimage yx yx yx

−4

16

28

16

0 + (−4)

0 + 16

−2 + 28

−2 + 16

Distance fromL(0, −2) toPreimage

Multiply Distanceby 4

Add Product toL(0, −2)

y

4

6

8

2

Ox

−24 6 82−2−4−6−8

−4

−6

−8

D

E

G

F

E′ F′

D′ G′

B′

A′

C′XB

C

A

T

−1

1

−1 −1

1

1

greater thanequal to

−1

0, −24

1

FF T

enVision™ Geometry • Teaching Resources


7-1 Additional PracticeDilations

1. Draw a dilation of ABCD with E as the center and with sides 1 __

2 as long.

2. What is the scale factor of the dilation shown?

For Exercises 3 and 4, find the coordinates of the vertices of each image.

3. D 0.75 (△ABC) , given A(4, −3), B(6, 1), C(10, −1)

4. D 1.5 (△XYZ) , given X(3, 0), Y(4, 2), Z(6, −2)

5. D k (△ABC) has a perimeter of 100 units and an area of 625 unit s 2 .

a. What is the perimeter of △ABC? b. What is the area of △ABC?

6. Charles enlarged the small kite MNOP to make a design for an art project, as shown.

a. How are the side lengths of the preimage and image related?

b. How are the areas related?

c. What is the scale factor of the dilation Charles used to enlarge the kite?

0.5

A’(3, −2.25), B’(4.5, 0.75), C’(7.5, −0.75)

X’(4.5, 0), Y’(6, 3), Z(9, −3)

The side lengths of the image are 3 times the corresponding side lengths of the preimage.

The area of the image is 9 times the area of the preimage.3

100 ____ k 625 ____

k 2

A

E

BC

DD′

C′ B′

A′

xO

y8

–4–4

–8

4 8–8

P

O

M

N

P′

O′

M′

N′

A′ B′

C′

D′

AB

C

D



7-2 Reteach to Build UnderstandingSimilarity Transformations

A similarity transformation is a composition of one or more rigid motions and a dilation. If a similarity transformation maps one figure to another, then the figures are similar.

1. Fill in the blanks.

A

B

C

A′

B′

C′

B′

Z

B″

A″

C″

A′

C′

△ABC is mapped to △A′B′C′ by a △A′B′C′ is mapped to △A″B″C″ by a . .

△ABC is to △A″B″C″.

2. Zachary says ( r 90° ◦ T <−5, −8> )(PQRS) is a similarity transformation. Which best explains why Zachary is incorrect?

𝖠 A dilation must be part of a similarity transformation.

𝖡 A rotation cannot be part of a similarity transformation.

𝖢 There cannot be two rigid motions in a similarity transformation.

𝖣 Quadrilaterals cannot be mapped to each other by a similarity transformation.

3. What are the vertices of ( D 2 ◦ T <−5, 2> )(△XYZ) for X(−1, −4), Y(2, 1), and Z(4, −3)?

X(−1, −4)

Y(2, 1)

Z(4, −3)

X′(−6, −2)

Y′(−3, 3)

Z′(−1, −1)

X″(−12, −4)

Y″(−6, 6)

Z″(−2, −2)

−1 + (−5)

2 + (−5)

4 + (−5)

−4 + 2

1 + 2

−3 + 2

�X″Y″Z″�XYZ �X′Y′Z′yx yx

−6(2)

−3(2)

−1(2)

−2(2)

3(2)

−1(2)

Translate by (−5, 2) Dilate by Scale Factor 2

reflection

similar

dilation



7-2 Additional PracticeSimilarity Transformations

What are the vertices of each image?

1. ( D 0.75 ∘ T ⟨−3, 2⟩ ) (△ABC) , given A(4, −3), B(6, 1), C(10, −1)

2. ( R x-axis ∘ r 270° ∘ D 2 ) (△XYZ), given X(6, 8), Y(3, 4), Z(5, −1)

3. ( T ⟨5, −2⟩ ∘ R y-axis ∘ D 0.5 ) (ABCD) , given A(2, 6), B(5, 7), C(8, 5), D(4, 2)

4. ( T ⟨−1, 4⟩ ∘ D (2,P) ) (△ABC), given A(−2, 1), B(2, 5), C(−2, 4), P(−4, 2)

Describe the similarity transformations and write the composition of transformations.

5.

xO

y

2

4

–2–2

–4

2A

B C

E D

B′ C′

E′

A′

D′

6.

xO

y

2

4

6

8

–2 2 4 6 8–4

C′

C′BB′

A

C

7. Luke says that the scale factor relating two figures is 0.6. Paula says the scale factor is 5 __

3 . If Paula is correct, explain why Luke is incorrect.

8. Carmen has a sign with dimensions 5 ft × 7.5 ft. She wants to reduce it to make a postcard. Postcard sizes are 3.5 in. × 5 in., 4 in. × 6 in., and 4.25 in. × 6 in. Which size postcard should she use? Explain.

A’(0.75, −0.75), B’(2.25, 2.25), C’(5.25, 0.75)

X’(16, 12), Y’(8, 6), Z’(−2, 10)

A’(4, 1), B’(2.5, 1.5), C’(1, 0.5), D’(3, −1)

A’(−1, 4), B’(7, 12), C’(−1, 10)

Dilation of 1 __ 3 centered at the

origin and a translation right 2. ( T ⟨2, 0⟩ ∘ D 1 _ 3

) (ABCDE) = A’B’C’D’E’

Answers may vary. Sample: Luke calculated the reciprocal of the true scale factor.

4 in. × 6 in.; 4 __ 5 = 6 ___

7.5 , which means the sign and the postcard

would be similar rectangles.

Rotation of 180° and a dilation of 2 centered at B(1, 3). ( D (2, B) ∘ r (180°, B ) ) (△ABC) = △A’B’C’



7-3 Reteach to Build UnderstandingProving Triangles Similar

A similarity transformation is a composition of one or more rigid motions and a dilation. If a similarity transformation maps one figure to another, then the figures are similar.

1. Match each pair of similar triangles to the appropriate theorem.

70°

70°33°

33°

12 15

6

8 10

4

16

20

20

25

108°108°

SSS Similarity If corresponding sides of two triangles are proportional, then the triangles are similar.

SAS Similarity If two pairs of corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar.

AA Similarity If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

2. Avery says the triangles shown are similar by the SAS Similarity Theorem. Which best explains why Avery’s answer is incorrect?

𝖠 The unknown side lengths are not proportional.

𝖡 The included angles are not congruent.

𝖢 The corresponding sides are not congruent.

𝖣 The unknown angle measures are not proportional.

3. Fill in the blanks to complete the proof.

Given: AB ⊥ BD EC ⊥ BD

Prove: △ABD ~ △ECD

Statement Reason

AB ⊥ BD and EC ⊥ BD Given∠ABD = 90° and ∠ECD = 90° Definition of perpendicular

∠ABD ≅ ∠ECD Transitive Property of Congruence∠ADB ≅ ∠EDC Reflexive Property of Congruence△ABD ~ △ECD AA ~ Theorem

66°

44°

3020 20

30

E

C B

A

D



7-3 Additional PracticeProving Triangles Similar

For Exercises 1–4, if the two triangles are similar, state why they are similar. If not, state that they are not similar.

1.

J L

M O

N

K

2.

N L

MO P

Q

30

15

13 1326 26

3. A

D

BE

C

4.

28

1014

5

T V

U

W

X

For Exercises 5 and 6, use the triangles shown.

5. What is FE?

6. What is DE?

For Exercises 7 and 8, what is the value of x?

7.

10 6

12

D

A C x

B

F

E 8.

3 21

14x

9. Are triangles ABC, DEF, and 10. The width of the pond shown HIJ similar? Explain. is x ft. What is the value of x?

4 9

1215

8

610

3 5

A H I

JF D

E

C

B

48 ft

60 ft

80 ft

A B

D

E

Cx ft

89° 89°35°

35°36

20 2440

RF

D

E

Q

P

not similar

not similar

43.2

x = 20

yes; by SSS ~x = 100 ft

48

SSS ~

SAS ~

x = 2



7-4 Reteach to Build UnderstandingSimilarity in Right Triangles

1. Use the diagram to match each description to the appropriate equation.

AB ___ AC

= AC ___ AD AB ___ CB

= CB ___ DB

AD ___ CD

= CD ___ DB

The length of the altitude is the geometric mean of the hypotenuse segments.

The length of the short leg is the geometric mean of the length of the hypotenuse and the length of the short segment.

The length of the long leg is the geometric mean of the length of the hypotenuse and the length of the long segment.

2. Given △JKL, Jamie writes a proportion to find MJ.

12 __ 18

= 18 ______ 12 + MJ

Jamie’s proportion is incorrect. Fill in the blanks to write the correct proportion.

12 __ 18

= 18 ___ MJ

3. Fill in the blanks and solve the proportion to find QP.

QR ____ QP

= QP ___

QS

4 + 28 _______ QP

= QP ___

4

(QP) 2 = 128

QP = 8 √ ___

2

AD hypotenuse

short leg

shortsegment

longsegment

BClong leg

altitude

L

K

12M

J

18

SQ

28

P R

4



7-4 Additional PracticeSimilarity in Right Triangles

1. Name the right triangles that are similar to △QRS.

S

Q

R

T

For Exercises 2–5, find the values of x and y.

2. 3.

4. 5.

6. Devin says that since the diagonals of the kite intersect at right angles, the small right triangles are similar to both the left half and the right half of the kite. Is he correct? Explain.

7. Isabel and Helena have built a frame and covered it with cloth. The frame is in the shape of a right triangle, △ABC, with side lengths 6 ft, 8 ft, and 10 ft. They use a vertical pole AE to raise corner A 3 ft, as shown. What is the distance ED from the base of the pole to the edge of the frame? Round to the nearest foot.

9

1

x

y

14

6

x y

12

4

x

3612 y

△QST, △SRT

x = 2 √ ___

30 ; y = 2 √ ___

21

x = 32 __ 3

Answers may vary. Sample: He is correct only if the vertical diagonal divides the kite into two congruent right triangles; otherwise he is incorrect.

4 ft

x = √ ___

10 ; y = 3 √ ___

10

y = 8 √ ___

2

C

D

B

A

E 10 ft

3 ft

8 ft

6 ft



7-5 Reteach to Build UnderstandingProportions in Triangles

1. Match each diagram to the appropriate conclusion.

A C

NM

B

A C

NM

B

A CM

B

Then...

AM ___ MB = CN ___ NB

Then...

AM ___ CM

= AB ___ CB

Then...

‾ MN ∥ ‾ AC

MN = 1 __ 2 (AC)

2. Marta incorrectly states that PS = SR. Fill in the blanks to explain Marta’s error and find the correct value of SR.

Marta assumes incorrectly that because QS is the of ∠PQR, S must be the midpoint of PR .

However, to find the correct value of SR, she should apply the Triangle- Angle-Bisector Theorem.

PS __ SR

= PQ ___

RQ

10 __ SR

= 12 __ 36

10 · = · SR

= SR

3. Fill in the blanks to find XJ.

XJ __ JY = ZK ___ KY

XJ __ 20

= 9 __ 12

· XJ =

XJ =

If... If... If...

Q

P RS

3612

10

Y

Z

KJ

X

12

9

20

angle bisector

36 30

180 15

12

12



7-5 Additional PracticeProportions in Triangles

Find the missing lengths. Round to the nearest tenth as needed. 1. x 2. y 3. z

Solve for x.

4. 2

6

9x

5. x + 1x

1512

6.

72x

8 16

7.

x 6

5 3

8.

x + 4x

x – 2 9

9.

7.52.5x

3x 4

10.

x + 4

x + 2x

3 11.

x

16 12

9

12. 6

9

x – 4

x

13. River claims that he can write two different proportions to find x. What are the two proportions?

14. The flag of Antigua and Barbuda is similar to the image shown, where DE || CF || BG.

a. Nora sketched the flag for a mural. The labels show the length of the lines in feet. What is the value of x?

b. What type of triangle is △ACF? Explain.

162.17.2

3

10

4

4

8

12

7

2

12

12 __ 16

= 9 __ x ; 12 __ 9 = 16 __ x

isosceles; because CA = FA

4

x

16 12

9

4x + 5

4x – 43x

21

D

C

B G

F

E

A

yz

x

7

3

5.6

8.75 7

25

4

5


Documents

Dilations - IT'S TRIMBLE TIME€¦ · Name PearsonRealize.com 7-1 Reteach to Build Understanding Dilations 1. The diagram shows a dilation of ABC with a scale factor of 2 and the