Geometry-Similar Figures ~1~...

Geometry-Similar Figures ~1~ NJCTL.org

Similar Figures Chapter Problems

Ratios and Proportions Class work Simplify the ratio.

1. 15 in to 45 in 2. 27yd to 6yd 3. 12 days to 4 weeks 4. 6 years to 1 decade

Solve the proportion

5.

6.

7.

8. x - 2

x -1=

1

2

Tell whether the statement is true or false.

9. If , then

10. If , then

11. If , then

12. The scale on the blueprint of a house is 0.04in = 1 foot. If the width of the kitchen on the blueprint is 1 inch, what is the actual width of the kitchen?

13. There are 350 people at the school basketball game. The ratio of the students to adults is 6:1. How many students attended the game?

Homework Simplify the ratio.

14. 40 feet to 12 feet 15. 8 days to 14 days 16. 150 feet to 1 mile 17. 20 ounces to 3 pounds

Solve the proportion.

18.


19.

20.

21.

Tell whether the statement is true or false.

22. If , then

23. If , then

24. If , then

25. Mike, Angela, and Victor have $160 in a ratio of 7:5:4. How much do they each have? 26. You made a 3-foot model of your home, using a scale of 1:42. What is the actual height of your

home? Similar Polygons using Transformations Classwork

Use the definition of similarity in terms of similarity transformations to determine if the two figures are similar. If similar, give the transformations in coordinate notation. 27.


28. 29.

30. 31.


Homework 32. 33.

34. 35.


36.

Similar Polygons using Corresponding Parts Classwork

37. Given that triangle XYZ ~ triangle LMN. a. Write as many congruence statements as possible about the sides and / or angles. b. Write the statement of proportionality. c. Write 5 more similarity statements.


38. The polygons below are similar. a. Write a similarity statement. b. What is the similarity ratio? c. What is the scale factor? d. List all congruent angles. e. Write the statement of proportionality.

39. Decide whether the polygons are similar. a. If yes, write a similarity statement. b. What is the similarity ratio? c. What is the scale factor?

In problems 40-41 DEFGHIJK, solve for the variables.

40.


108°

135°35°

A

E

D

B

C 35°

135°

108°

I

H

J

F

G

41.

Homework

42. Given that triangle PQR ~ triangle DEF. a. Write as many congruence statements as possible about the sides and / or angles. b. Write the statement of proportionality. c. Write 5 more similarity statements.

43. The polygons below are similar. a. Write a similarity statement. b. What is the similarity ratio? c. What is the scale factor? d. List all congruent angles. e. Write the statement of proportionality.

In problems 44-45, decide whether the polygons are similar.

a. If yes, write a similarity statement. b. What is the similarity ratio? c. What is the scale factor?


44.

45.

46. ABCDPQRS. Solve for the variables.

Similar Triangles Classwork

47. Determine if the triangles are similar. If so, state the similarity postulate or theorem. a.


b.

c.

48. Given: DH ||FG

Prove: DDEH : DGEF

49. Given: ÐR @ ÐA , PR = 9 , QR = 7.2 , AB = 4.8 , and AC = 6

Prove: DQPR : DBCA

E

H

D

G

F


50. Given: ÐA @ ÐD, ÐB @ ÐE

Prove: DABC : DDEF using similarity transformations

51. Given: FD

CA

EF

BC

DE

AB

Prove: DABC : DDEF using similarity transformations

Homework

52. Determine if the triangles are similar. If so, state the similarity postulate or theorem. a.

b.

A

B

C D

E

F

A

B

C D

E

F


c.

53. Given: BD || AE

Prove: DACE : DBCD

54. Given: mÐP = 48°, mÐQ = 55°, mÐB = 55°, mÐC = 77°

Prove: DPQR : DABC

55. Given: FD

CA

DE

AB , ÐA @ ÐD

Prove: DABC : DDEFusing similarity transformations

A

B

C D

E

F


Proportions in Similar Triangles Classwork

In problems 56-58, determine if DE || BC .

56.

57.

58.


Solve for y. 59.

60.

61.

62.

63.


A E

C

B D

64. ABC is mapped to ADE under a dilation with a scale factor of 3, explain why BC is parallel to DE .

65. Prove the Side Splitter Theorem.

Given BD ║ AE

Prove CE

CD

CA

CB

6

4

2

5A

D

E

C

B


Homework

In problems 66-68, determine if DE || BC . 66.

67.

68.

Solve for y.

69.


70.

71.

72.

73.


A E

C

B D

74. ABC is mapped to ADE under a dilation with a scale factor of 1.5, explain why BC is parallel to DE .

75. Prove the Converse to the Side Splitter Theorem.

Given DC

ED

BC

AB

Prove BD ║ AE

6

4

2

5A

D

E

C

B


Similar Circles Class work

76. Describe the similarity transformations needed to map circle A to circle A’. Point A is the center of the dilation. a. Find the constant of dilation. b. Identify the translation vector.

77. Which similarity transformations can map circle A with center (0,0) and radius 2 to circle B with center (-2, 3) and radius 4. Point A is the center of the dilation.

78. Which similarity transformations can map circle A with center (-3, -5) and radius 2 to circle B

with center (-6, 7) and radius 3. Point A is the center of the dilation.



80. Prove all circles are similar. Given circle A with radius x and circle B with radius y Prove circle A is similar to circle B

Homework

81. Describe the similarity transformations needed to map circle A to circle A’. Point A is the center of the dilation.

a. Find the constant of dilation. b. Identify the translation vector.

yxA

B


82. Describe the similarity transformations needed to map circle A with radius AB to circle A with

radius AB '. Point A is the center of the dilation. a. Find the constant of dilation. b. Identify the translation vector.

83. Which similarity transformations can map circle A with center ( 3,3) and radius 5 to circle B with center (-3, 3) and radius 4. Point A is the center of the dilation.

84. Which similarity transformations can map circle A with center (-3, -3) and radius 4 to circle B

with center (-3, -3) and radius 5. Point A is the center of the dilation.


Solve Problems using Similarity Class work

86. A basketball hoop in your backyard casts a shadow 109 inches long. You are 5 feet 8 inches tall and cast a shadow 62 inches long. Find the height of the basketball hoop in inches. Round your answer to the nearest whole number.

87. You want to know the approximate height of a very tall pine tree. You place a mirror on the ground and stand where you can see the top of the tree in the mirror. How tall is the tree? The mirror is 24 feet from the base of the tree. You are 24 inches from the mirror and your eyes are 6 feet above the ground. Round your answer to the nearest tenth.


88. To find the distance d across a lake, you locate the points as shown. Find d. Round your answer to the nearest tenth.

89. A graphic designer wants to design a new grid system for a poster. The poster is 27 inches by 36

inches. The grid must have margins of 2 inch along all edges. There must be 4 rows of rectangles. The rectangles must be similar in size to the poster.

a. What should be the height of the rectangles? b. What should be the width of the rectangles? c. How many columns of rectangles can there be?

Homework

90. A yardstick casts a shadow 1 ft long. A nearby tree casts a 16 ft shadow. How tall is the tree? Round your answer to the nearest tenth.

91. You want to know the approximate height of a tall oak tree. You place a mirror on the ground and stand where you can see the top of the tree in the mirror. How tall is the tree? The mirror is 24 feet from the base of the tree. You are 36 inches from the mirror and your eyes are 5 feet above the ground. Round your answer to the nearest tenth.

92. To find the distance d across a lake, you locate the points as shown. Find d. Round your answer

to the nearest tenth.

120 ft

15 ft

20 ft

d

15 ft

56 ft

28 ft

d


93. A graphic designer wants to design a new grid system for a poster. The poster is 27 inches by 36 inches. The grid must have margins of 1 inch along all edges. There must be 5 rows of rectangles. The rectangles must be similar in size to the poster.

a. What should be the height of the rectangles? b. What should be the width of the rectangles? c. How many columns of rectangles can there be?


Similar Figures Unit Review Multiple Choice - Choose the correct answer for each question. No partial credit will be given.

1. Simplify the ratio 15 inches to 3 inches. a. 15 to 9 b. 1:5 c. 5/3 d. 5/1

2. Solve the proportion.

a. 30 b. 54 c. 24 d. 27

3. Solve the proportion.

a. x = 5 b. x = -5 c. x = 1 d. x = 0.5

4. Use the definition of similarity, C is the center of dilation.

a. Not Similar b. Translation (x,y) -> (x-1, y+3) followed by constant of dilation=2 c. Constant of dilation=2 followed by Translation (x,y) -> (x-1, y+3) d. b and c order doesn’t matter

10

8

6

4

2

5

D''

C'' B''D

C B


5. Decide whether the triangles are similar. If so, write a similarity statement.

a. Yes, DABC : DDEF

b. Yes, DABC : DDFE

c. Yes, DABC : DFDE d. The triangles are not similar

In problems 6-7, JKLMPQRS, Find x.

6.

a. 4 b. 6.67 c. 2.5 d. 3.75

7.

a. 55 b. 114 c. 56 d. 135


8. Describe the similarity transformation needed to map ABC to A’B’C’ using coordinate

notation.

a. Not Similar b. (x, y) -> (1x/4, 1y/4) c. (x, y) -> (4x, 4y) d. (x, y) -> (6x, 6y)

9. What is the similarity ratio r needed to map ABC to A’B’C’? What is the scale factor f?

a. r=4, f=4 b. r=1/4, f=1/4 c. r=4, f=1/4 d. r=1/4, f=4

8

6

4

2

5

C'B'

A

CB

8

6

4

2

5

C'B'

A

CB


10. Determine if the triangles are similar. If so, state the similarity postulate or theorem.

a. Yes, by AA

b. Yes, by SSS

c. Yes, by SAS

d. The triangles are not similar

11.

a. 3.6 b. 4.4 c. 40 d. 48

12. Solve for y

a. 8 b. 4.5 c. 6 d. 12

13. Which similarity transformations can map circle A with center (-8,-3) and radius 6 to circle B

with center (4,-2) and radius 3. Point A is the center of the dilation. a. (x, y) -> (x + 12, y +1) constant of dilation=2 b. (x, y) -> (x + 4, y + 1) constant of dilation=1/2 c. (x, y) -> (x + 12, y - 1) constant of dilation=1/2 d. None of the above


Short Constructed Response - Write the answer for each question. No partial credit will be given. 14. The scale on a map of the US, 1 inch = 250 miles. New York is 12 inches from California. What is the actual distance between the cities.

15. For the diagram below, ADE is mapped to ABC under a dilation with a scale factor of 1/3, explain why

BC is parallel to DE .

16. Your school casts a shadow 30 feet long. At the same time a person 6 feet casts a shadow 4 feet

long. Sketch and label a diagram. Find the height of your school. Round your answer to the nearest tenth.

6

4

2

5A

D

E

C

B


4

2

5A

D

EC

B

Extended Constructed Response - Solve the problem, showing all work. Partial credit may be given.

17. Given ECDB ,

Prove triangle ABC ~ triangle ADE using similarity transformations


Answer Key

1. 1/3 2. 9/2 3. 3/7 4. 3/5 5. 2.67 6. 5.25 7. 5 8. 5 9. true 10. true 11. false 12. 25 feet 13. 300 students 14. 10/3 15. 4/7 16. 5/176 17. 5/12 18. 2 19. 3 20. 3 21. -1 22. true 23. false 24. true 25. Mike - $70, Angela - $50, Victor - $40 26. 126 feet 27. (x,y) ->(x+4,y+1) 28. (x,y)->(x/2,y/2) 29. (x,y)->(2x,2y) 30. (x,y)->(x,-y) (x,y)->((x/2,y/2) 31. not similar 32. (x,y)->(x/2,y/2) 33. (x,y)->(3x/2,3y/2) 34. not similar 35. (x,y)->((x/2,y/2) (x,y)->(x,y+2.5)) 36. (x,y)->(-y,x) 37. a. ÐX @ ÐL , ÐY @ ÐM ,

ÐZ @ ÐN

b. XY/LM=YZ/MN=XZ/LN c. XZY~LNM,YXZ~MLN, YZX~MNL,ZXY~NLM, ZYX~NML 38. a. ABCD~EFGH b. r=2/3 c. f=3/2 d. <A=<E, <B=<F, <C=<G, <D=<H e. AB/EF=BC/FG= CD/GH=DA/HE 39. not similar 40. w=12, x=28, y=2.4, z=2.67 41. w=8, x=3, y=4.5 42. a. ÐP @ ÐD, ÐQ @ ÐE,

ÐR @ ÐF b. PQ/DE=QR/EF=PR/DF c. PRQ~DFE, QPR~EDF, QRP~EFD, RPQ~FDE, RQP~FED 43. a. ABCDE~HIJFG b. not enough info c. not enough info d. <A=<H, <B=<I, <C=<J, <D=<F, <E=<G e. AB/HI=BC/IJ=CD/JF=DE/FG=EA/GH 44. a. ABCD~QPSR b. r=2/1 c. f=1/2 45. not similar 46. w=114,x=4.5, y=2.25,z=87 47. a. not similar b. yes by SAS

c. not similar 48. <D = <G and <E = <E so by

AA 49. (QR/BA)=(PR/AC) and

<R=<A so by SAS 50. see below 51. see below 52.

a. yes, by AA or SAS

b. not similar

c. yes, by SSS

53. <B=<A and <D=<E so by

AA 54. <Q=<B and <R=<C so by

AA~ 55. see below 56. yes 57. no 58. no 59. 12 60. 10.67 61. 7 62. 11.36 63. 8.75 64. ABC~ADE because a dilation is a similarity transformation. <B=<D because corresponding angles of ~ triangles are congruent. BC is parallel to DE by the corresponding angles converse. 65. see below 66. yes 67. yes 68. no 69. 14 70. 10 71. 11.25 72. 4 73. 10


74. ABC~ADE because a dilation is a similarity transformation. <B=<D because corresponding angles of ~ triangles are congruent. BC is parallel to DE by the corresponding angles converse. 75. see below 76. a. constant of dilation=1/3 b. vector AA’ = <7, -3> 77. constant of dilation=2 (x,y)->(x-2, y+3) 78. constant of dilation=3/2 (x,y)->(x-3, y+12) 79. constant of dilation=1/2 (x,y)->(x-6, y+3) 80. see below 81. a. constant of dilation=3/2 (x,y) -> (x+3, y+5) b. vector AB = <3, 5> 82.

a. constant of dilation=2 b. vector AA’ = <0, 0> 83. constant of dilation=5/4 (x,y)->(x-6, y) 84. constant of dilation=5/4 (x,y)->(x,y) 85. constant of dilation=1 (x,y)->(x-6, y+3) 86. 120 inches 87. 72 ft 88. 90 ft 89. a. 8 in or 5.75 in b. 6 in or 7.67 in c. 3 or 4 90. 48 ft 91. 40 ft 92. 45 ft 93. a. 6.8 in or 5 in b. 5.1 in or 6.67 in c. 4 or 5 Unit Review

1. D 2. C 3. A 4. D 5. C 6. D 7. A 8. B 9. C 10. C 11. A 12. A 13. D 14. 3000 miles 15. ADE~ABC because a dilation is a similarity transformation. <D=<B because corresponding angles of ~ triangles are congruent. BC is parallel to DE by the corresponding angles converse. 16. 45 feet 17. see below

Proofs

50. AA~ proof using transformations

<A=<D,<B=<E Given

Dilate ABC with sf=DE/AB Def of scale factor

ABC~A'B'C' Def of dilation

<A=<A',<B=<B' corr angles of ~ triangles are cong

A'B'=(DE/AB)*AB=DE simplify

<A'=<D,<B'=<E Transitive Prop of congruence

A'B'C'=DEF ASA congruence

A'B'C' ~ DEF Def of congruence

ABC ~ DEF Transitive prop of ~

51. SSS~ proof using transformations

AB/DE=BC/EF=CA/FD Given


DE/AB=EF/BC=FD/CA Definition of Proportions

Dilate ABC with scale factor k =DE/AB Definition of Scale Factor

ABC~A'B'C' Definition of Dilation

A'B'=(DE/AB)(AB)=DE Simplify

B'C'=(EF/BC)(BC)=EF Substitution / Simplify

C'A'=(FD/CA)(CA)=FD Substitution / Simplify

A'B'C' @ DEF SSS=

A'B'C'~DEF Definition of @

ABC~DEF Transitive Property of ~

55. SAS~ proof using transformations

AB/DE=CA/FD Given

<A=<D Given

DE/AB=FD/CA Definition of Proportions

Dilate ABC with scale factor k =DE/AB Definition of Scale Factor

ABC~A'B'C' Definition of Dilation

A'B'=(DE/AB)(AB)=DE Simplify

C'A'=(FD/CA)(CA)=FD Substition / Simplify

A'B'C' @ DEF SAS @

A'B'C'~DEF Definition of @

ABC~DEF Transitive Property of ~

65. Side Splitter Theorem Proof

EA parallel to DB Given

<CBD=<CAE corresponding angles postulate

<C=<C reflexive prop of congruence

CBD ~ CAE AA~

CA/CB=CE/CD corr sides of ~ triangles are prop

CB+BA=CA, CD+DE=CE segment addition postulate

(CB+BA)/CB=(CD+DE)/CD substitution

CB/CB+BA/CB=CD/CD+DE/CD simplify

1+BA/CB=1+DE/CD simplify

BA/CB=DE/CD subtraction prop of =

CB/BA=CD/DE property of proportions

75. Converse of Side Splitter Theorem Proof

AB/BC=ED/DC Given

1+AB/BC=1+ED/DC Addition property of =

BC/BC+AB/BC=DC/DC+ED/DC substitution

(BC+AB)/BC=(DC+ED)/DC simplify

BC+AB=AC,DC+ED=CE segment addition postulate

AC/BC=CE/DC substitution


<C=<C reflexive property of congruence

BCD ~ ACE SAS~

<CBD = <CAE corresponding angles of ~ triangles are congruent

BD is parallel to AE corresponding angles converse

80. Prove all circles are similar

Translate circle A with vector AB getting circle A' Definition of Translation

circle A is congruent to circle A' Definition of Translation

center of circle A' is B Definition of Translation

Dilate circle A' with scale factor k = y/x Definition of Dilation

circle A' ~ circle B Definition of Dilation

circle A ~ circle B Transitive Property of ~

Unit Review #17 AA~ proof using transformations

<B=<D,<C=<E Given

Dilate ABC with sf=DE/BC Def of scale factor

ABC~A'B'C' Def of dilation

<B=<B',<C=<C' corr angles of ~ triangles are cong

B'C'=DE/BC*BC=DE simplify

<B'=<D,<C'=<E Transitive Prop of congruence

A'B'C'=ADE ASA congruence

A'B'C' ~ ADE Def of congruence

ABC ~ ADE Transitive prop of ~

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