Discovering Geometry An Investigative Approach

Answers to Exercises CHAPTER 8 • CHAPTER

8

CHAPTER 8 • CHAPTER

LESSON 8.1

1. 228 m 2 2. 41.85 cm 2 3. 8 yd 4. 21 cm 5. 91 ft 2 6. 182 m 2 7. 96 in2 8. 210 cm 9. A 42 ft 2 10. sample answer: 6 cm

12 cm 48 cm2

Answers to Exercises

4 cm

8 cm

11. 12. 13. 14.

48 cm2

3 square units 10 square units 712 square units sample answers: 8 cm

16 cm 64 cm2

23. 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:

64 cm2

4 cm

8 cm

15. possible answer: 16 cm 16 cm

4 cm

A

30°

M

30°

16 cm

16 cm

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 m 2. 1 19. 530 20. 112 21. 96 square units 22. 32 square units

96

ANSWERS TO EXERCISES

29a.

29c.

29b.

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:

.) 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 1 (a b)(a b). If you add the areas of the 2 three triangles, the area of the trapezoid is 1 c 2 ab. 2 22. A b1 B h D

9 cm 12 cm

9 cm

14. sample answer: 4 cm

7 cm

8 cm

7 cm

10 cm

9 cm

15. sample answer:

46 cm

12 cm

45 cm

35 cm

12 cm

56 cm

16. The length of the base of the triangle equals the sum of the lengths of both bases of the trapezoid.

C

Given: trapezoid ABCD with height h. area of ABD 21 hb1; area of BCD 21hb2; area of trapezoid sum of areas of two triangles 21hb1 b2 23. 1141 square units 24. 7 square units 25. 70 m 26. 144 cm2 27. 828 ft 2; 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 112° by the Inscribed Angle of the circle. mBC by the Central Conjecture, and d e mBC Angle Conjecture. OBA is congruent to OCA is a semicircle, so by SSS, so d e 56°. DEC mDE 68°. By the Inscribed Angle Conjecture, p 34°. Using OEC and the Triangle Sum Conjecture, n 90°. 31. 32.623.6.3.6

2 cm 3 cm 7 cm

3 cm 9 cm


97


12 cm

b2

LESSON 8.3


1a. 121,952 ft 2 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 ft 2 more surface area 4. 21 5. 336 ft2; $1780

98


6. $760 7. 220 terra cotta tiles, 1107 blue tiles; $1598.15 8. 72 cm 2 9. AB 16.5 cm, BD 15.3 cm 10. 60 cm2 by either method 11. Because AOB is isosceles, mA 20° and 82°. mAOB 140°. mA A B 140° and mCD mBD because parallel lines intercept mAC 360° 140° 82° 69°. congruent arcs on a circle. 2 12. E

11. a2 2ab b2

USING YOUR ALGEBRA SKILLS 8

1. x 2 6x 5

a

b

a

a2

ab

a

b

ab

b2

b

2. 2x 2 7x

x5 x

x

12. a2 b2 b

a2

ab

ab

b 2

5 x

x2

x1 x

a

13. (x 15)(x 4)

1

2x 7 x

x

15

x

x2

15x

4

4x

60

14. (x 12)(x 2) x

12

x

x2

12x

2

2x

24

7

15. (x 5)(x 4)

3. 6x 2 19x 10 3x 2 x

x

x

x

5

x2

5x

4x

20

2 x

x 4 2x 5

16. (x 3)2 (x 3)(x 3)

x

x

4. (3)(2x 1)

5. (x 5)(x 3) x5

3 x

3

3

x2

3x

3x

9


5

x

17. (x 6)(x 6)

5

x

6

x2

6x

6x

36

x 2x 1

x3

x2

x

x

3

x

6

18. (2x 7)(2x 7)

1

6. (2x 3)(x 4) x

x4

2x 3 x

3

x

4

7. x 2 26x 165 x

15

x2

15x

11x

165

x 11

9. x 2 8x 16 x

4

x

x2

4x

4

4x

16

8. 12x 2 13x 35 3x

7

4x

12x 2

28x

5

15x

35

10. 4x 2 25 2x

5

2x

4x 2

10x

5

10x

25

2x

7

2x

4x 2

14x

7

14x

49

19. x 4 or x 1 20. x 10 or x 3 21. x 3 or x 8 1 22. x 2 or x 4 h 23a and b. h h4

1 23c. 2[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.


99

LESSON 8.4

2092 cm2 2. 74 cm 256 cm 4. 33 cm2 63 cm 6. 490 cm2 57.6 m 8. 25 ft 42 cm2 10. 58 cm2 1 1 1 1 11. a 2s; A 2asn 2 2s s 4 s 2 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.


1. 3. 5. 7. 9.

17. Area is 36 square units. y

y – _4 x 12 3 y – _1 x 6 3

(6, 4)

x

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 31 the height of the whole triangle. Each has an area 61 the area of the whole triangle. By the same argument, the other small triangles also have areas 61 the area of the whole triangle.

y y _12x 5 1

(2, 6) y 2x 10

x

100


h

2

19. nw ny 2x 20. 504 cm2 21. 840 cm2

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.

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. 2 24° 48° 20. x mDE 21. 90° 38° 28° 28° 180° 22.

24 cm

r 18 cm

12 cm

804 m2 11,310 km2 154 m2 4 times

18 cm


12. 13. 14. 15.

6 cm


101

LESSON 8.6


1. 6 cm2 64 2. 3 cm2 3. 192 cm2 4. ( 2) cm2 5. (48 32) cm2 6. 33 cm2 7. 21 cm2 105 8. 2 cm2 9. 6 cm 10. 7 cm 11. 75 12. 100 13. 42 14. $448 15a.

15b.

15c.

102


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 90 20. True. If 24 360 2r, then r 48 cm. 360 21. True. If n 24, then n 15. 22. False. It could be a rhombus. 23. true; Triangle Inequality Conjecture

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 2 9. 414.7 cm 10. 329.1 cm2 11. area of square 4 area of trapezoid 4 area of triangle 12. $1570 13. sample answer:

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


14. sample tiling

33.42/32.4.3.4/44


103

CHAPTER 8 REVIEW

1. B (parallelogram) 3. C (trapezoid) 5. F (regular polygon) 7. J (sector) 9. G (cylinder) 11.

2. A (triangle) 4. E (kite) 6. D (circle) 8. I (annulus) 10. H (cone) 12.

20. 23. 26. 29. 32.

32 cm 21. 32 cm 81 cm2 24. 48 cm 153.9 cm2 27. 72 cm2 300 cm2 30. 940 cm2 Area is 112 square units.

22. 25. 28. 31.

15 cm 40° 30.9 cm2 1356 cm2

y

Apothem

D (6, 8)

A (0, 0)

C (20, 8)

B (14, 0)

x

33. Area is 81 square units.

13.

y R (4, 15)

U (9, 5)


F (0, 0)

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. b

O (4, –3)

34. 6 cm 35. 172.5 cm2 36. sample answers:

b

h

h

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 b2h. b2

b1

h

h b1

b2

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 r 2. ThustheareaofacircleisgivenbytheformulaAr 2. r ␲r

18. 5990.4 cm2 17. 800 cm2 19. 60 cm2 or about 188.5 cm2 104

x


37. 1250 m2 38. Circle. For the square, 100 4s, s 25, A 252 625 ft2. For the circle, 100 2r, 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. 2r1 44 cm. 2r2 22 cm, which implies 4r2 44 cm. Therefore r1 2r2. The area of the large bundle is 4r22 cm2. The combined area of two small bundles is 2r22 cm2. Thus he is getting half as much for the same price. 47. $2002 48. $384 (16 gal)