Answers to ALL Ch4 WS - Wikispaces to ALL Ch4_WS 4.1 Practice A 1. JK;3,2 ...

Answers to ALL Ch4_WS

4.1 Practice A

1. ; 3, 2JK −

2. (3, 1), (0, 2), (1, 3)A B C′ ′ ′− − −

3. 4, 6−

4. ( , ) ( 6, 4)x y x y→ + +

5. (6, 6)Q′ 6. ( 2, 11)M ′ − −

7. 8.

9. no; Sample answer: The translation from A to A′could be different than the translation from A′ to

.A′′

10. , ) ( 2, 3);( y x yx → − − 0.25 mi

4.1 Practice B

1. ( 4, 1), (1, 7), ( 1, 3)F G H′ ′ ′− −

2. 11, 17−

3. , ) ( 5, 6)( y x yx → + −

4. ( 6, 7)G′ − 5. ( 14, 8)H ′ −

6.

7. no; Multiplying x by 2 does not simply move ortranslate the object, it stretches the shape.

8. ( , ) ( 3, 1); nox y x y→ + −

9. Sample answer:

yes; All the vectors are parallel. This makes sense because the vertices are all translated by the same vector, so the segments joining the vertices to their images all have the same slope. Because they have the same slope, they are parallel.

x

y4

−4

2−4 −2

A

C

B

B′

C′

A′

x

y4

−4

−2

2 4−4 −2

E

C

DD′

C′

E′

x

y

8

−8

−4

4 8 12−4

EC

D

D′

C′

C″

D″

E″

E′

x

y8

−8

84−4−8

G

FH

F′ H′

G′

x

y

4

2

−2

4−2

A

BC

A′

B′C′

6

6

L″

K″

x

y

−8

−4

−4

K

J

LL′

J′

K′

12

−12

J″

Answers

4.2 Practice A

1. 2.

3. 4.

5. 6.

x

y4

−4

42−2−4

A

C

B

B′

C′

A′

x

y

4

−8

8−8A

BB′

A′

CC′

x

y8

4

−8

8−4−8 A

B

B′

A′

−12

C

C′

x

y

3

1

−3

−5

31−1−5

J′

J″

J

K′K″

K

L′

L″

L

x

y

4

−8

8−8

S

P

S′P′

Q′

Q

R

R′

x

y8

−8

8

R

R′

S′S

P

P′

Q′

Q

Answers 7.

8. x-axis, y-axis, any line joining opposite outer pointsor opposite inner points of the star

9. ,x y= y-axis, any line joining an outer point to its

opposite inner point of the star

10. (0, 1) 11. 1a =

12. no; Two translations can always be written as asingle translation, so the process is the same as aglide reflection.

4.2 Practice B

1.

2.

3. 4.

5. 6.

7. x-axis, y-axis, ,y x= and y x= −

8. (1, 0)P

9. yes; Sample answer: You can reflect it twice in thesame line.

10. no; Angle measures and distances remain constantunder any rigid motion.

11. yes; Two reflections do not combine into anotherreflection, so the motion is not a glide reflection.

4.2 Enrichment and Extension

1. ( , ) and ( , )a c a b− −

rectangle

2. ( , ) and ( , )a c a b− −

3.

isosceles trapezoid

4.

isosceles right triangle

x

y8

−4

4−8

J′J″

K′K″

KL″ J

L′

L

x

y4

2

4−4

CC′

DD′

EE′

x

y8

4

−8

−4

8

C C′

D

E E′

16

D′

x

y8

−8

−4NN′M

M′

L

L′

K

K′x

y8

4

−4

N

N′

L

L′

K

K′

MM′

x

y

2

6

−2

4−2

A″

AB″

BB′

C″

CC′

6

A′

x

y

2

6

−2

4−2

A″A

B″

BB′

C″

CC′

A′

x

y2

−4

2−2

F(a, c)F′(−a, c)

G(a, b)G′(−a, b)

x

y

2

−4

−2

2−2−4

B

B′

A

A′

x

y

2

4

−2

2−2−4

B

A

A′

Answers 5. 6. 4.3 Practice A

1. 2.

3.

4.

5. yes; 45 , 90 , 135 , 180° ° ° ° 6. yes; 30 , 120° °

7.

square

8. H, I, N, O, S, X, Z; H: 180 ; I: 180 ; N: 180 ;° ° °O: all angles; S: 180 ; X: 90 , 180 ; Z: 180° ° ° °

4.3 Practice B

1.

x

y

−2

2−2−4

A′ A″

B′B″

C′ C″

x

y

4

6

8

2

−2−4−6

A′

A″B′

B″ C′

C″A

A′

B

B′

C

C′

D

D′

Px

y4

−4

4−4

J

J′L

L′

K

K′

MM′

x

y

4

−2

4

R″

RS″

S

S′

T″

T

T′

R′

x

y4

−4

42−2−4

A(2, 0)

B(0, 2)

(−2, 0)

(0, −2)

x

y

−4

4

T T′

U U′ V

V′W

W′

x

y

4

2−4

R″

R

S″

SS′

T″

TT′

R′

Answers 2.

3.

4. yes; 90 , 180 , 270° ° ° 5. yes; 180°

6. no; A 360° rotation just takes the shape back to theoriginal, so it must be the same.

7. yes; A rotation of 180° maps the figure onto itself;6 times

8. no 9. yes

10. 0, 8; 0:180 ; 8: 180 ;° ° Some students may also

include the number 1, depending on how it iswritten.

4.3 Enrichment Extension

1. 2 4

3 3y x= − + 2. 8y x= − +

3. 2

33

y x= + 4. 1

52

y x= − +

5. a. 0 0 0( , )x y a x+ −

b. 0 0 0 0( , )x b y a x y− + − +

c. (2, 7)−

d. (9, 12)

6. a.

x

y4

−4

4 6 82

C″

C

C′

D″

D

D′

E″

EE′

x

y

4

6

2

42−2−4

F″

F′

G″G′

x

y4

4

C″

D″

D

D′

E″

E

E′

C

C′

4.4 Practice A

1. Triangle 1, triangle 5, and triangle 8 are allcongruent to one another; Square 6 is congruent tosquare 2; Parallelogram 4 is congruent toparallelogram 7.

2. Triangle 1, triangle 2, and triangle 9 are all congruentto one another; Rectangle 3 is congruent to rectangle10; Triangle 5 is congruent to triangle 8.

3. translation 8 units right and 2 units up

4. reflection in the x-axis, followed by a translation2 units right

5. congruent; ABCΔ can be mapped on to STUΔ bya reflection in the y-axis, followed by a translation1 unit right and 7 units down.

6. not congruent; It is not possible to transformpolygon EFGH to polygon WXYZ using only rigidtransformations.

7. , ,CD CD DE D E′′ ′′ ′′ and ;CE C E′′ ′′

EE DD CC′′ ′′ ′′≅ ≅

8. 14° 9. 45°

4.4 Practice B

1. Squares 1, 3, and 7 are congruent; Triangles 6 and 8are congruent; Parallelograms 4 and 5 arecongruent; Each can be translated to one of theothers by a sequence of rigid motions.

2. not congruent; Although three of the four pointsretain their relative positions, the fourth pointchanges the shape, so the quadrilaterals are notcongruent.

3. 8 units 4. 160°

5. no; The angle between lines a and b is not 90 .°

6. 180°

7. sometimes; It depends on the shape of the objectundergoing the transformation.

8.

9. yes; It will produce a translation of the same imageafter any even number of reflections. This is nottrue for odd numbers of reflections, which willresult in a reflection of the original image.

4.4 Enrichment and Extension

1. 7

1

−

2. 1 5 7 1

3 3 1 1

− −

3. 6 1

9 10

− −

4. 5 15 5

0 11 4

−

5. 1 5 3

1 0 1

−

6. 1 1 1

3 3 3

− − −

7. 1 1 1 1 5 3 0 4 2

3 3 3 1 0 1 4 3 2

− − − + = −

8.

The coordinates of triangle A B C′ ′ ′ are the sameas the image matrix in Exercise 7.

A″

A′A

st

x

y

2

−2

62

B

C

C′A

B′A′

Answers

4.5 Practice A

1. 3; enlargement 2. 2.5; reduction

3. 4.

5. 6.

7.

8. 0.77

9. the old film-style camera

10. no; Every dimension would dilate by the same scalefactor k, so the area would increase by 2 ,k onefactor of k for each dimension.

11. no; A scale factor of 1 does not dilate the object atall. The object is neither enlarged nor reduced.

4.5 Practice B

1. 16; reduction 2. 1.5; enlargement

3.

4. 5.

6.

7. It would look like it is 80 millimeters across.

8. A dilation with a scale factor of 0k = would sendall the vertices to the center of the dilation, so theobject would be reduced to a point.

x

y

B,

CP

D

AA′

D′ C′

B′

x

y

B

C

P

D

AA′

D′ C′

B′

x

y

B

C,

P

D

A

A′

D′ C′

B′

x

y

4

−8

8−4

Q

R

PP′ Q′

R′S′

S

P B,

C

D

A

A′

B′

C′

D′

P B

C

D

AA′

B′

C′D′

x

y4

2

−4

−2

2

M

M′ LL′

KK′

J

J′

x

y

−12

8 12−4

X

X′

Y

Y′

Z

Z′

WV

V′

W′

x

y8

−8

84−4−8

AB

DA′

B′

C′

D′

C

Answers 9. yes; The perimeter is additive, so it is scaled by the

same factor by which the object is dilated.

10. The scale factor is x; 2x =

4.5 Enrichment and Extension

1.

2. The length and width double; The perimeterdoubles; The area increases by a factor of 4.

3. The perimeter is 4 times as large, and the area is16 times as large.

4. The perimeter increases by a factor of a, and thearea increases by a factor of 2.a

5. ( , ) (2 , 2 )x y x y→ 6. 223.04 mm

4.6 Practice A

1. 2.

3. reflection in the y-axis, followed by a dilation witha scale factor of 2

4. yes; The triangle is a translation;( , ) ( 5, 1)x y x y→ + − followed by a dilation of

( )2 23 3

( , ) , ;x y x y→ Points C and F do not follow

these transformations, so it is not a similaritytransformation.

5. yes; The quadrilateral can first be rotated 180°about the origin (or, reflected in the y-axis and thenthe x-axis). Then the figure can be dilated with ascale factor of 0.5k = and translated to its finalposition.

6. Rotate PQRΔ so that side a is parallel to side b.

Translate GHIΔ so that point G maps to point P.Because translations preserve angle measure, andall of the angles of an equilateral triangle are 60 ,°

GHIΔ lies on .PQRΔ Because, GI

coincides with

PR

and GH

coincides with ,PQ GI

lies on PR

and GH lies on .PQ Finally, dilate PQRΔ about

point P by a scale factor of b

a so that it is the same

size as .GHIΔ Because a similarity transformationmaps PQRΔ onto ,GHIΔ the triangles are similar.

x

y

8

4

84 1612

Length Width Perimeter Area

1. Points A, B,C, and D

2 4 12 8

2. Points ,A B′ ′, and C D′ ′

4 8 24 32

3. Points D, E,F, and G

8 16 48 128

x

y

4

2−2−4−6

R

Q

P

R′R″

Q″

P″

Q′P′

x

y

−8

−4

8−4R

Q

P″

P

R′ R″

Q″P′

Q′

Answers 7. no; A square and a rectangle are not similar, so you

cannot use a similarity transformation to change theshape of the object.

8. no; For example begin with a unit square centeredat the origin. If you perform a dilation centered atthe origin with a scale factor 2 and then translate1 unit right, the result is not the same as if you firsttranslate the square 1 unit right and then performa dilation centered at the origin with a scale factorof 2.

9. All white triangles are dilations and translations.There are no rotations in the image.

4.6 Practice B

1.

2.

3. a 180° rotation followed by a dilation with a scalefactor of 3

4. similar; The transformation is a translation 6 unitsto the right and 3 units up, followed by a dilationwith a scale factor 2.

5. not similar; The transformation is a reflection in they-axis, followed by a translation, however twovertices were translated 2 units up and two verticeswere translated 3 units up.

6. Rotate onto ABE DBCΔ Δ such that ABE∠coincides with .DBC∠ Because rotations preserve

length and measure, ABE DBC∠ ≅ ∠ and AE is

still parallel to .DC So, AB coincides with BD

and BE coincides with .BC Dilate ABEΔ until

AE coincides with .CD Therefore, ABEΔ issimilar to .DBCΔ

7. no; Circles of different size are simply a dilation ofeach other, so they remain similar.

8. no; The edges are distorted and curved, and arenot an exact replica of the original text. So, amagnifying glass does not produce a perfectsimilarity transformation because the imageis distorted.

9. no; Similar triangles do not need to be the samesize, so there are more similar triangles than thereare congruent triangles.

4.6 Enrichment and Extension

1. 15 square units 2. 4

3. 4 square units 4. 64 square units

5. The area has increased by a factor of 24 16.=

6. a. 1 23 33, ,k a b= = = −

b. Because k is 3, the radius of Circle A is 3 timesthe radius of Circle B, so 3 .t r=

Cumulative Review

1. 5 4x − 2. 7 11x + 3. 4 15x +

4. 9 99x− − 5. 4 8x − 6. 6 36x− +

7. 3x 8. 9 6x − 9. 3 6x− +

10. 9x− + 11. 5 88x − 12. 20 74x −

13. 32x = 14. 10x = − 15. 12x = −

16. 23x = 17. 30x = 18. 8x = −

19. 96x = − 20. 8x = 21. 3x =

22. 22x = − 23. 4x = − 24. 3x = −

25. 20x = − 26. 2x = 27. 8x = −

28. 12x = −

29. a. $9.35

b. $10.65

30. a. 48 fluid ounces

b. 6 cups

c. 3 pints

d. 1.5 quarts

x

y4

2

−4

−2

4−4

E

E′

D

D′

D″

C

C′

C″

E″

x

y8

4

−8

−4

8 12EE′

D″C″

E″

D

D′

C

C′