LESSON Practice 9.1 For use with pages 598–605vanguardhighschoolgeometry.yolasite.com/resources/Geo_Workshee… · 7 s 6G 22. Offi ce Supplies Two offi ces submit supply Offi

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114GeometryChapter 9 Resource Book

LESSON

9.1 PracticeFor use with pages 598–605

Use the translation (x, y) → (x 1 6, y 2 3).

1. What is the image of A(3, 2)? 2. What is the image of B(24, 1)?

3. What is the preimage of C9(2, 27)? 4. What is the preimage of D9(23, 22)?

The vertices of n ABC are A(21, 1), B(4, 21), and C(2, 4). Graph the image of the triangle using prime notation.

5. (x, y) → (x 2 3, y 1 5) 6. (x, y) → (x 2 4, y 2 2)

x

y

2

2

x

y

1

1

n A9B9C9 is the image of n ABC after a translation. Write a rule for the translation. Then verify that the translation is an isometry.

7.

x

y

1

1

A

B

C

A9

B9

C9

8.

x

y

1

1

A

B

C

A9

B9

C9

Name the vector and write its component form.

9.

J

M

10.

X

Y

11. D

R

Use the point P(5, 22). Find the component form of the vector that describes the translation to P9.

12. P9(2, 0) 13. P9(8, 23) 14. P9(0, 4) 15. P9(25, 24)

LAHGE11FLPW_c09_114-128.indd 114 2/12/09 1:30:03 PM

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115Geometry

Chapter 9 Resource Book

The vertices of n ABC are A(1, 2), B(2, 6), and C(3, 1). Translate n ABC using the given vector. Graph n ABC and its image.

16. ⟨8, 2⟩ 17. ⟨27, 23⟩

x

y

2

2

x

y

2

2

Find the value of each variable in the translation.

18.

x

y

1008

808 813

2ba8

c

5d 8

19.

x

y

318

2012

3c 1 2

b 2 5

a8

20. Navigation A hot air balloon is fl ying from

x

yD(14, 12)

C(8, 8)

B(6, 3)

A(0, 0)

N

point A to point D. After the balloon travels 6 miles east and 3 miles north, the wind direction changes at point B. The balloon travels to point C as shown in the diagram.

a. Write the component form for ### Y AB and ### Y BC .

b. The wind direction changes and the balloon travels from point C to point D. Write the component form for ### Y CD .

c. What is the total distance the balloon travels?

d. Suppose the balloon went straight from A to D. Write the component form of the vector that describes this path. What is this distance?

LESSON

9.1 Practice continuedFor use with pages 598–605


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LESSON


Use the diagram to write a matrix

x

y

2

1

A

B

FE

D

Cto represent the polygon.

1. n CDE

2. n ABF

3. Quadrilateral BCEF

4. Hexagon ABCDEF

Add or subtract.

5. f6 3g 1 f1 9g 6. F28 4 4 25

G 1 F 4 6 6 21G

7. F 5 22 2 4 27 2

G 1 F 1 3 6 24 6 21

G 8. f20.3 1.8g 2 f0.6 2.7g

9. F21 29 0 2

G 2 F 5 9 26 27G 10. F 1.4 1.3

25 26.5 2 4

G 2 F 21.4 23 3.9 4 1.3 3.9

GFind the image matrix that represents the translation of the polygon. Then graph the polygon and its image.

A B C M N O P

11. F 21 5 3 2 2 6G; 5 units right and 12. F 3 7 5 1

1 2 6 5G; 6 units left and 3 units down 2 units up

x

y

4

2 x

y

2

2

Multiply.

13. f4 23gF 26 2G 14. f20.8 4gF 3

21.6G 15. F22 3 5 24

GF21 4 7 5

G 16. F 0.9 5

24 2GF 3 0 24 23G 17. f23 2 6gF 25

0 23G

18. F 2 5 5 1 0 3GF 0

24 2G


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117Geometry


Use the described translation and the graph of the image to fi nd the matrix that represents the preimage.

19. 3 units right and 4 units up 20. 2 units left and 3 units down

x

y

1

1

A9

B9

C9

D9

x

y

1

1

A9

B9

C9

D9

E9

21. Matrix Equation Use the description of a translation of a triangle to fi nd the value of each variable. What are the coordinates of the vertices of the image triangle?

F 28 x 28 4 4 yG 1 F 22 b c

d 25 2G 5 F r 24 23 7 s 6G

22. Offi ce Supplies Two offi ces submit supply Offi ce 1

15 weekly planners

5 chair mats

20 desk trays

Offi ce 2

25 weekly planners

6 chair mats

30 desk trays

lists. A weekly planner costs $8, a chairmat costs $90, and a desk tray costs $5. Use matrix multiplication to fi nd the total cost of supplies for each offi ce.

23. School Play The school play was performed on three evenings. The attendance on each evening is shown in the table. Adult tickets sold for $5 and student tickets sold for $3.50.

Night Adults Students

First 340 250

Second 425 360

Third 440 390

a. Use matrix addition to fi nd the total number of people that attended each night of the school play.

b. Use matrix multiplication to fi nd how much money was collected from all tickets each night.

LESSON



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LESSON


Graph the refl ection of the polygon in the given line.

1. x-axis 2. y-axis 3. x 5 21

x

y

1

1

A

B

C

x

y

1

1A D

B C

x

y4

1

A

B

C

Graph the refl ection of the polygon in the given line. Use the distance formula to show the fi gure and image are congruent.

4. y 5 1 5. y 5 2x 6. y 5 x

x

y

1

1

A D

B C

x

y

1

3

AD

BC

x

y

1

1

A

B

C

Use matrix multiplication to fi nd the image. Graph the polygon and its image.

A B C A B C D

7. Refl ect F 23 1 6 4 7 2G in the x-axis. 8. Refl ect F 2 5 7 1

6 4 25 23G in the y-axis.

x

y

2

2 x

y

2

2


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119Geometry


Write a matrix for the polygon. Then fi nd the image matrix that represents the polygon after a refl ection in the given line.

9. x-axis 10. y-axis 11. x-axis

x

y

1

1

A

B

C

x

y

1

1

A

B

C

D

x

y

1

2

A

BC

Find point C on the x-axis so AC 1 BC is a minimum.

12. A(2, 22), B(11, 24) 13. A(21, 4), B(6, 3) 14. A(23, 2), B(26, 24)

The vertices of n ABC are A(22, 1), B(3, 4), and C(3, 1). Refl ect n ABC in the fi rst line. Then refl ect n A9B9C9 in the second line. Graph n A9B9C9 and n A0B 0C 0.

15. In y 5 1, then in y 5 22 16. In x 5 4, then in y 5 21 17. In y 5 x, then in x 5 22

x

y

2

2

x

y

2

2

x

y

2

4

18. Laying Cable Underground electrical cable is being laid for two new homes. Where along the road (line m) should the transformer box be placed so that there is a minimum distance from the box to each of the homes?

LESSON


LAHGE11FLPW_c09_114-128.indd 119LAHGE11FLPW_c09_114-128.indd 119 11/17/09 10:42:08 AM11/17/09 10:42:08 AM

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LESSON


Match the diagram with the angle of rotation.

1.

x8

2.

x8

3.x8

A. 1108 B. 1708 C. 508

Trace the polygon and point P on paper. Then draw a rotation of the polygon the given number of degrees about P.

4. 458 5. 1208 6. 1358

PA

B

C

P

A B

C

P

A B

C

D

Rotate the fi gure the given number of degrees about the origin. List the coordinates of the vertices of the image. Show that the fi gure and image are congruent.

7. 908 8. 1808 9. 2708

x

y

2

1

A D

CB

x

y

1

1A

D

C

B

x

y

1

1

AD

CB

Find the value of each variable in the rotation.

10.

y

x

x 2 411

1308

11. y

x2x 1 36

1008

12.

r

4s

2s 2 324

2708


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121Geometry


Find the image matrix that represents the rotation of the polygon about the origin. Then graph the polygon and its image.

A B C A B C

13. F 1 4 3 2 2 4G; 908 14. F 0 4 2

21 0 3G; 1808

x

y

1

1

x

y

2

4

A B C D A B C D

15. F 1 2 4 5 21 3 3 21G; 908 16. F 23 22 2 1

24 21 21 24G; 2708

x

y

1

2

x

y

1

1

The endpoints of } CD are C(2, 1) and D(4, 5). Graph }

C9D9 and } C 0D 0 after the given rotations.

17. Rotation: 908 about the origin 18. Rotation: 1808 about the origin Rotation: 2708 about (2, 0) Rotation: 908 about (0, 23)

x

y

4

2

x

y2

2

LESSON



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LESSON


The endpoints of } CD are C(1, 2) and D(5, 4). Graph the image of } CD after the glide refl ection.

1. Translation: (x, y) → (x 2 4, y) 2. Translation: (x, y) → (x, y 1 2) Refl ection: in the x-axis Refl ection: in y 5 x

x

y

1

1

x

y

1

1

The vertices of n ABC are A(3, 1), B(1, 5), and C(5, 3). Graph the image of n ABC after a composition of the transformations in the order they are listed.

3. Translation: (x, y) → (x 1 3, y 2 5) 4. Translation: (x, y) → (x 2 6, y 1 1) Refl ection: in the y-axis Rotation: 908 about the origin

x

y

1

21

x

y1

21

Graph } F 0G 0 after a composition of the transformations in the order they are listed. Then perform the transformations in reverse order. Does the order affect the fi nal image } F 0G 0 ?

5. F(4, 24), G(1, 22) 6. F(21, 23), G(24, 22)

Rotation: 908 about the origin Refl ection: in the line x 5 1 Refl ection: in the y-axis Translation: (x, y) → (x 1 2, y 1 10)

x

y

1

21

x

y

1

1


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123Geometry


Verify that the fi gures are congruent by describing the composition of transformations.

7.

x

y

1

5

AA99

A9 B9

C9D9

D99

C99

B99

B

CD

8.

x

y

1

1

A

A99

A9

B9

C9

B99

C99B

C

In the diagram, k i m, } AB is refl ected in line k, and }

A9B9 is refl ected in line m.

9. A translation maps }

AB onto which segment?

A

A9

A99

B

B9

B99

k

m

10. Which lines are perpendicular to @###$ BB0 ?

11. Name two segments parallel to }

AA0 .

12. If the distance between k and m is 2.7 centimeters, what is the length of

} AA0 ?

13. Is the distance from A9 to m the same as the distance from A0 to m? Explain.

Find the angle of rotation that maps A onto A0.

14.

A

A9

A99

k

m

608

15.

AA9

A99

k

m

458

16. Stenciling a Border The border pattern below was made with a stencil. Describe how the border was created using one stencil four times.

LESSON



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FOCUS ON


Does the shape tessellate? If so, tell whether the tessellation is regular.

1. Right triangle 2. Irregular hexagon 3. Parallelogram

Use the steps in Example 2 to make a fi gure that will tessellate.

4. Make a tessellation using a square as the base fi gure.

5. Make a tessellation using a hexagon as the base fi gure. Change one pair of opposite sides.

6. Make a tessellation using a trapezoid as the base fi gure. Change both pairs of opposite sides.

Verify that a tessellation can be made using the given polygons.

7. 8. 9.

Describe the transformation(s) used to make the tessellation.

10. 11.

12. 13.

14. Challenge Tessellations occur often in the real world, especially in nature. A bee’s honeycomb is a tessellation of hexagons. A brick wall is a tessellation of rectangles. Think of one example of a real-world tessellation and draw it.


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125Geometry


LESSON


Determine whether the fi gure has rotational symmetry. If so, describe the rotations that map the fi gure onto itself.

1. 2. 3. 4.

Does the fi gure have the rotational symmetry shown? If not, does the fi gure have any rotational symmetry?

5. 1208 6. 1808 7. 458

8. 368 9. 1808 10. 908

In Exercises 11–16, draw a fi gure for the description. If not possible, write not possible.

11. A triangle with exactly two lines 12. A quadrilateral with exactly two lines of symmetry of symmetry

13. A pentagon with exactly two lines 14. A hexagon with exactly two lines of symmetry of symmetry


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15. An octagon with exactly two lines 16. A quadrilateral with exactly four lines of symmetry of symmetry

17. Paper Folding A piece of paper is folded in half and some cuts are made, as shown. Which fi gure represents the piece of paper unfolded?

A. B. C. D.

In Exercises 18 and 19, use the following information.

Taj Mahal The Taj Mahal, located in India, was built between 1631 and 1653 by the emperor Shah Jahan as a monument to his wife. The fl oor map of the Taj Mahal is shown.

18. How many lines of symmetry does the fl oor map have?

19. Does the fl oor map have rotational symmetry? If so, describe a rotation that maps the pattern onto itself.

In Exercises 20 and 21, use the following information.

Drains Refer to the diagram below of a drain in a sink.

20. Does the drain have rotational symmetry? If so, describe the rotations that map the image onto itself.

21. Would your answer to Exercise 20 change if you disregard the shading of the fi gures? Explain your reasoning.

Practice continuedFor use with pages 645–650

LESSON

9.6


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127Geometry


LESSON


Find the scale factor. Tell whether the dilation is a reduction or an enlargement. Then fi nd the values of the variables.

1.

6

4

12x

y

5P 9

P C 2.

x5

6

12P 9

P

C

Use the origin as the center of the dilation and the given scale factor to fi nd the coordinates of the vertices of the image of the polygon.

3. k 5 3 4. k 5 1 }

3

x

y

1

1

M

L

N

x

y

2

2

I

H

G

5. k 5 2 6. k 5 5 }

2

x

y

1

1

AC

B

D

x

y

1

1

P

S

R

A dilation maps A to A9 and B to B9. Find the scale factor of the dilation. Find the center of the dilation.

7. A(4, 2), A9(5, 1), B(10, 6), B9(8, 3)

8. A(1, 6), A9(3, 2), B(2, 12), B9(6, 20)

9. A(3, 6), A9(6, 3), B(11, 10), B9(8, 4)

10. A(24, 1), A9(25, 3), B(21, 0), B9(1, 1)


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The vertices of ~ABCD are A(1, 1), B(3, 5), C(11, 5), and D(9, 1). Graph the image of the parallelogram after a composition of the transformations in the order they are listed.

11. Translation: (x, y) → (x 1 5, y 2 2)

Dilation: centered at the origin with a scale factor of 3 } 5

x

y

1

2

12. Dilation: centered at the origin with a scale factor of 2

Refl ection: in the x-axis

x

y

2224

In Exercises 13–15, use the following information.

Flashlight Image You are projecting images onto a wall with a fl ashlight. The lamp of the fl ashlight is 8.3 centimeters away from the wall. The preimage is imprinted onto a clear cap that fi ts over the end of the fl ashlight. This cap has a diameter of 3 centimeters. The preimage has a height of 2 centimeters and the lamp of the fl ashlight is located 2.7 centimeters from the preimage.

13. Sketch a diagram of the dilation.

14. Find the diameter of the circle of light projected onto the wall from the fl ashlight.

15. Find the height of the image projected onto the wall.

LESSON



Documents

LESSON Practice 9.1 For use with pages 598–605vanguardhighschoolgeometry.yolasite.com/resources/Geo_Workshee… · 7 s 6G 22. Offi ce Supplies Two offi ces submit supply Offi