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CHAPTER 8 • CHAPTER CHAPTER 8 • CHAPTER

LESSON 8.1

1. 228 m2

2. 41.85 cm2

3. 8 yd

4. 21 cm

5. 91 ft2

6. 182 m2

7. 96 in2

8. 210 cm

9. A � 42 ft2

10. sample answer:

11. 3 square units

12. 10 square units

13. 7�12

� square units

14. sample answers:

15. possible answer:

16. 23.1 m2

17. 2(4)(3) � 2(5.5)(3) � 57 m2

18. For a constant perimeter, area is maximized by

a square. 100 m � 4 � 25 m per side; A � 625 m2.

19. �5

130�

20. 112

21. 96 square units

22. 32 square units

4 cm

16 cm16 cm

16 cm16 cm

64 cm2

8 cm

4 cm

16 cm

64 cm2

8 cm

4 cm

12 cm

8 cm

6 cm

48 cm2

48 cm2

Answers to Exercises23. 500 cm2

24a. smallest: 191.88 cm2; largest: 194.68 cm2

24b. Answers will vary. Sample answer: about

193 cm2.

24c. Answers will vary. The smallest and largest

area values differ at the ones place, so the digits

after the decimal point are insignificant compared

to the effect of the limit of precision in the

measurements.

25a. In one Ohio Star block, the sum of the red

patches is 36 in2, the sum of the blue patches is

72 in2, and the yellow patch is 36 in2.

25b. 42

25c. About 1814 in2 of red fabric, about 3629 in2

of blue fabric, and about 1814 in2 of yellow

fabric. The border requires 5580 in2 (if it does

not need the extra 20%).

26. 100; 36 � 64. The area of the square on the

longer side is the same as the sum of the areas on

the other two legs.

27. a � 76°,b � 52°,c � 104°,d � 52°,e � 76°,

f � 47°, g � 90°, h � 43°, k � 104°, m � 86°.

Explanations will vary.

28. sample construction:

29a. 29b.

29c.

30°

30°A M

8

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LESSON 8.2

1. 20 cm2

2. 49.5 m2

3. 300 square units

4. 60 cm2

5. 6 cm

6. 9 ft

7. 30 ft

8. 5 cm

9. 16 m

10. 168 cm

11. 12 cm

12. 3.6 ft; 10.8 ft

13. sample answer:

14. sample answer:

15. sample answer:

16. The length of the base of the triangle equals

the sum of the lengths of both bases of the

trapezoid.

2 cm

3 cm 3 cm

7 cm 9 cm

12 cm

46 cm 35 cm

45 cm 56 cm

12 cm

4 cm

8 cm

10 cm 9 cm

7 cm

7 cm

12 cm

9 cm 12 cm

9 cm

17. �12

�(To see why, draw altitude PQ�.)

18. more than half, because the top card

completely covers one corner of the bottom card

19a. 86 in. of balsa wood and 960 in2 of Mylar

19b. 56 in. (or less, if he tilts the kite)

20. 3600 shingles (to cover an area of 900 ft2)

21. The isosceles triangle is a right triangle

because the angles on either side of the right

angle are complementary. If you use the trapezoid

area formula, the area of the trapezoid is �12

�(a � b)(a � b). If you add the areas of the

three triangles, the area of the trapezoid is�12

�c2 � ab.

22.

Given: trapezoid ABCD with height h. area

of �ABD � �12

� hb1; area of �BCD � �12

�hb2; area

of trapezoid � sum of areas of two triangles

� �12

�h�b1 � b2�23. 11�

14

� square units

24. 7 square units

25. 70 m

26. 144 cm2

27. 828 ft2; 144 ft

28. 1440 cm2; 220 cm

29a. incenter

29b. orthocenter

29c. centroid

30. a � 34°, b � 68°, c � 68°, d � 56°, e � 56°,

f � 90°, g � 34°, h � 56°, m � 56°, n � 90°,

p � 34°. Possible explanation: Let O be the center

of the circle. mBC� � 112° by the Inscribed Angle

Conjecture, and d � e � mBC� by the Central

Angle Conjecture. �OBA is congruent to �OCA

by SSS, so d � e � 56°. DEC� is a semicircle, so

mDE� � 68°. By the Inscribed Angle Conjecture,

p � 34°. Using �OEC and the Triangle Sum

Conjecture, n � 90°.

31. 32.62�3.6.3.6

D

A b1

b2

h

B

C

LESSON 8.3

1a. 121,952 ft2

1b. 244 gal of base paint and 488 gal of finishing

paint

2. He should buy at least four rolls of wallpaper.

(The area of each roll is 125 ft2. The total surface

area to be papered is 480 ft2.) If paper cut off at the

corners is wasted, he’ll need 5 rolls.

3. 1552 ft2; 776 ft2 more surface area

4. 21

5. 336 ft2; $1780

6. $760

7. 220 terra cotta tiles, 1107 blue tiles; $1598.15

8. 72 cm2

9. AB � 16.5 cm, BD � 15.3 cm

10. 60 cm2 by either method

11. Because �AOB is isosceles, m�A � 20° and

m�AOB � 140°. mAAAB� � 140° and mCD� � 82°.

mAC�� mBD� because parallel lines intercept

congruent arcs on a circle.�360°�1240°� 82°�� 69°.

12. E

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ANSWERS TO EXERCISES 99

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USING YOUR ALGEBRA SKILLS 8

1. x2 � 6x � 5 2. 2x2 � 7x

3. 6x2 � 19x � 10

4. (3)(2x � 1) 5. (x � 5)(x � 3)

6. (2x � 3)(x � 4)

7. x2 � 26x � 165 8. 12x2 � 13x � 35

9. x2 � 8x � 16 10. 4x2 � 25

2x

�5

52x

4x 2 10x

�10x �25

x

4

4x

x 2 4x

4x 16

4x

5

�73x

12x 2 �28x

15x �35

x

�11

�15x

x 2 �15x

�11x 165

x � 4x

4

x2x � 3

x 3

x � 3x

3

5x

x � 5

x 2

2x � 1

x

x

3

1

2x � 5

x

x

5

x3x � 2x x 2

2x � 7

x

x

7

x

x � 1 x

1

5xx � 5

x 2

11. a2 � 2ab � b2 12. a2 � b2

13. (x � 15)(x � 4) 14. (x � 12)(x � 2)

15. (x � 5)(x � 4)

16. (x � 3)2 � (x � 3)(x � 3)

17. (x � 6)(x � 6)

18. (2x � 7)(2x � 7)

19. x � �4 or x � �1

20. x � �10 or x � 3

21. x � 3 or x � 8

22. x � ��12� or x � �4

23a and b.

23c. �12

�[h � (h � 4)]h � 48

23d. h � �8 or h � 6. The height cannot be

negative, so the only valid solution is h � 6.

The height is 6 feet, one base is 6 feet, and the

other base is 10 feet.

h � 4

h

h

2x

�7

72x

4x 2 14x

�14x �49

x

�6

6x

x 2 6x

�6x �36

x

�3

�3x

x 2 �3x

�3x 9

x

�4

5x

x 2 5x

�4x �20

x

2

�12x

x 2 �12x

2x �24

x

4

15x

x 2 15x

4x 60

a

�b

ba

a 2 ab

�ab �b 2

a

b

ba

a 2 ab

ab b 2

LESSON 8.4

1. 2092 cm2 2. 74 cm

3. 256 cm 4. 33 cm2

5. 63 cm 6. 490 cm2

7. � 57.6 m 8. � 25 ft

9. � 42 cm2 10. � 58 cm2

11. a � �12

�s; A � �12

�asn � �12

� � �12

�s � s � 4 � s2

12. It is impossible to increase its area, because a

regular pentagon maximizes the area. Any

dragging of the vertices decreases the area.

(Subsequent dragging to space them out more

evenly can increase the area again, but never

beyond that of the regular pentagon.)

13. � 996 cm2

14. � 497 cm2

15. total surface area � 13,680 in2 � 95 ft2;

cost � $8075

16. Area is 20 square units.

x

y

y � x � 5 1_2

y � �2x � 10

(2, 6)

17. Area is 36 square units.

18. Conjecture: The three medians of a triangle

divide the triangle into six triangles of equal area.

Argument: Triangles 1 and 2 have equal area

because they have equal bases and the same

height. Because the centroid divides each median

into thirds, you can show that the height of

triangles 1 and 2 is �13

� the height of the whole

triangle. Each has an area �16

� the area of the whole

triangle. By the same argument, the other small

triangles also have areas �16

� the area of the whole

triangle.

19. nw � ny � 2x

20. 504 cm2

21. 840 cm2

h12

x

y

y � – x � 12 4_3

y � – x � 6 1_3(6, 4)

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ANSWERS TO EXERCISES 101

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LESSON 8.5

1. 9� in2

2. 49� cm2

3. 0.8 m2

4. 3 cm

5. �3� in.

6. 0.5 m

7. 36� in2

8. 7846 m2

9. 25� � 48, or about 30.5 square units

10. 100� � 128, or about 186 square units

11.

12. 804 m2

13. 11,310 km2

14. 154 m2

15. 4 times

r � 18 cm

16. A � �r 2 because the 100-gon almost

completely fills the circle.

17. 456 cm2

18. 36 ft2

19. The triangles have equal area when the point

is at the intersection of the two diagonals. There is

no other location at which all four triangles have

equal area.

20. x � mDE� � 2 � 24° � 48°

21. 90° � 38° � 28° � 28° � 180°

22.

18 cm6 cm

12 cm

24 cm

LESSON 8.6

1. 6� cm2

2. �64

3�� cm2

3. 192� cm2

4. (� � 2) cm2

5. (48� � 32) cm2

6. 33� cm2

7. 21� cm2

8. �10

25�� cm2

9. 6 cm

10. 7 cm

11. 75

12. 100

13. 42

14. $448

15a.

15b.

15c.

15d.

16. sample answer:

17a. (144 � 36�) cm2; 78.54%

17b. (144 � 36�) cm2; 78.54%

17c. (144 � 36�) cm2; 78.54%

17d. (144 � 36�) cm2; 78.54%

18. 480 m2

19. AB � 17.0 cm, AG � 6.6 cm

20. True. If 24� � �39600

� � 2�r, then r � 48 cm.

21. True. If �3

n60� � 24, then n � 15.

22. False. It could be a rhombus.

23. true; Triangle Inequality Conjecture

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ANSWERS TO EXERCISES 103

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LESSON 8.7

1. 150 cm2 2. 4070 cm2

3. 216 cm2 4. 340 cm2

5. � 103.7 cm2 6. � 1187.5 cm2

7. � 1604.4 cm2 8. � 1040 cm2

9. � 414.7 cm2 10. � 329.1 cm2

11. area of square � 4 � area of trapezoid � 4 �area of triangle

12. $1570

13. sample answer:

14. sample tiling

33.42/32.4.3.4/44

15. a � 75°, b � 75°, c � 30°, d � 60°,

e � 150°, f � 30°

16. About 23 days. Each sector is about 1.767 km2.

17. a � 50°, b � 50°, c � 80°, d � 100°, e � 80°,

f � 100°, g � 80°, h � 80°, k � 80°, m � 20°,

n � 80°. Explanations will vary. Sample

explanation: The angle with measure d corresponds

to the angle forming a linear pair with g. Because

d � 100°, by the Parallel Lines Conjecture, the

angle adjacent to g measures 100°, and by the

Linear Pair Conjecture, g � 80°. The angle with

measure f corresponds to the angle measuring

100°, so f � 100°. The angles measuring g and k

are the base angles of an isosceles triangle, so by

the Isosceles Triangle Conjecture, k � 80°.

18. 398 square units

CHAPTER 8 REVIEW

1. B (parallelogram) 2. A (triangle)

3. C (trapezoid) 4. E (kite)

5. F (regular polygon) 6. D (circle)

7. J (sector) 8. I (annulus)

9. G (cylinder) 10. H (cone)

11. 12.

13.

14. Sample answer: Construct an altitude from

the vertex of an obtuse angle to the base. Cut off

the right triangle and move it to the opposite side,

forming a rectangle. Because the parallelogram’s

area hasn’t changed, its area equals the area of the

rectangle. Because the area of the rectangle is

given by the formula A � bh, the area of the

parallelogram is also given by A � bh.

15. Sample answer: Make a copy of the trapezoid

and put the two copies together to form a

parallelogram with base �b1 � b2� and height h.

Thus the area of one trapezoid is given by the

formula A � �12

� �b1 � b2�h.

16. Sample answer:Cut a circular region into a large

number of wedges and arrange them into a shape

that resembles a rectangle.The base length of this

“rectangle”is�r and the height is r, so its area is�r2.

ThustheareaofacircleisgivenbytheformulaA��r2.

17. 800 cm2 18. 5990.4 cm2

19. 60� cm2 or about 188.5 cm2

�r

r

b1

b2

b2

b1

h h

bh

bh

Apothem

20. 32 cm 21. 32 cm 22. 15 cm

23. 81� cm2 24. 48� cm 25. 40°

26. 153.9 cm2 27. 72 cm2 28. 30.9 cm2

29. 300 cm2 30. 940 cm2 31. 1356 cm2

32. Area is 112 square units.

33. Area is 81 square units.

34. 6 cm 35. 172.5 cm2

36. sample answers:

37. 1250 m2

38. Circle. For the square, 100 � 4s, s � 25,

A � 252 � 625 ft2. For the circle, 100 � 2�r,

r � 15.9, A � �(15.9)2 � 794 ft2.

39. A round peg in a square hole is a better fit.

The round peg fills about 78.5% of area of the

square hole, whereas the square peg fills only about

63.7% of the area of the round hole.

40. giant 41. about 14 oz

42. One-eighth of a 12-inch diameter pie;

one-fourth of a 6-inch pie and one-eighth of a

12-inch pie both have the same length of crust,

which is longer than one-sixth of an 8-inch pie.

43a. 96 ft; 40 ft 43b. 3290 ft2

44. $3000 45. $4160

46. It’s a bad deal.2�r1 � 44 cm.2�r2 � 22 cm,

which implies 4�r2 � 44 cm. Therefore r1 � 2r2.

The area of the large bundle is 4��r2�2 cm2. The

combined area of two small bundles is 2��r2�2 cm2.

Thus he is getting half as much for the same price.

47. $2002 48. $384 (16 gal)

x

y

R (4, 15)

U (9, 5)

O (4, –3)

F (0, 0)

xA (0, 0)

D (6, 8) C (20, 8)

y

B (14, 0)

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