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Perimeters and Areas of Similar Figures

4 9. 16 81. perimeters: ; areas:. Perimeters and Areas of Similar Figures. Lesson 10-4. Lesson Quiz. 1. For the similar rectangles, give the ratios (smaller to larger) of the perimeters and of the areas.

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Perimeters and Areas of Similar Figures

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  1. 4 9 16 81 perimeters: ; areas: Perimeters and Areas of Similar Figures Lesson 10-4 Lesson Quiz 1. For the similar rectangles, give the ratios (smaller to larger) of the perimeters and of the areas. 2. The triangles are similar. The area of the largertriangle is 48 ft2. Find the area of the smaller triangle. 3. The similarity ratio of two regular octagons is 5 : 9. The area of the smaller octagon is 100 in.2 Find the area of the larger octagon. 4. The areas of two equilateral triangles are 27 yd2 and 75 yd2. Find their similarity ratio and the ratio of their perimeters. 5. Mulch to cover an 8-ft by 16-ft rectangular garden costs $48. At the same rate, what would be the cost of mulch to cover a 12-ft by 24-ft rectangular garden? 27 ft2 324 in.2 3 : 5; 3 : 5 $108 10-5

  2. 1. The perimeter is 4(6) = 24 m. The area A of a regular polygon is half the apothem a times the perimeter p: A = ap = (3)(24) = 36 m2 2. The perimeter is 6(42) = 252 in. The area A of a regular polygon is half the apothem a times the perimeter p: A = ap = (36)(252) = 4536 in.2 3. The perimeter is 6(8) = 48 ft. The area A of a regular polygon is half the apothem a times the perimeter p: A = ap = (7)(48) = 168 ft2 1 2 1 2 1 2 1 2 1 2 1 2 Trigonometry and Area Lesson 10-5 Check Skills You’ll Need Solutions 10-5

  3. Trigonometry and Area Lesson 10-5 Check Skills You’ll Need (For help, go to Lesson 10-3.) Find the area of each regular polygon. 1. 2. 3. 36 m2 168 ft2 4536 in2 Check Skills You’ll Need 10-5

  4. Trigonometry and Area Lesson 10-5 Notes 10-5

  5. Trigonometry and Area Lesson 10-5 Notes 10-5

  6. Find the perimeter p and apothem a, and then find the area using the formula A = ap. 1 2 360 10 Because the polygon has 10 sides, mACB = = 36. and are radii, so CA = CB. Therefore, ACMBCM by the HL Theorem, so CA CB 1 2 1 2 m ACM = m ACB = 18 and AM = AB = 6. Trigonometry and Area Lesson 10-5 Additional Examples Finding Area Find the area of a regular polygon with 10 sides and side length 12 cm. Because the polygon has 10 sides and each side is 12 cm long, p = 10 • 12 = 120 cm. Use trigonometry to find a. 10-5

  7. Use the tangent ratio. 6 a tan 18° = 6 tan 18° a = Solve for a. Substitute for a and p. 1 2 A = ap Simplify. 360 18 1107.966073 Use a calculator. 6 . tan 18° 1 2 360 tan 18° A = • • 120 A = Trigonometry and Area Lesson 10-5 Additional Examples (continued) Now substitute into the area formula. The area is about 1108 cm2. Quick Check 10-5

  8. Find the perimeter p and apothem a, and then find the area using the formula A = ap. 1 2 360 5 Because the pentagon has 5 sides, mACB = = 72. CA and CB are radii, so CA = CB. Therefore, ACMBCM by the HL Theorem, so 1 2 m ACM = m ACB = 36 Trigonometry and Area Lesson 10-5 Additional Examples Real-World Connection The radius of a garden in the shape of a regular pentagon is 18 feet. Find the area of the garden. 10-5

  9. Use the cosine ratio to find a. Use the sine ratio to find AM. AM 18 a 18 sin 36° = cos 36° = Use the ratio. a = 18(cos 36°) Solve. AM = 18(sin 36°) Use AM to find p. Because ACMBCM, AB = 2 • AM. Because the pentagon is regular, p = 5 • AB. Trigonometry and Area Lesson 10-5 Additional Examples (continued) So p = 5 • (2 • AM) = 10 • AM = 10 • 18(sin 36°) = 180(sin 36°). 10-5

  10. 1 2 A = • 18(cos 36°) • 180(sin 36°) Substitute for a and p. A = 1620(cos 36°) • (sin 36°) Simplify. Use a calculator. A770.355778 Trigonometry and Area Lesson 10-5 Additional Examples (continued) 1 2 Finally, substitute into the area formula A = ap. The area of the garden is about 770 ft2. Quick Check 10-5

  11. A triangular park has two sides that measure 200 ft and 300 ft and form a 65° angle. Find the area of the park to the nearest hundred square feet. Theorem 9-1 1 2 Area = • side length • side length• sine of included angle Substitute. 1 2 Area = • 200 • 300 • sin 65° Area = 30,000 sin 65° Simplify. Use a calculator 27189.23361 Trigonometry and Area Lesson 10-5 Additional Examples Real-World Connection Use Theorem 9-1: The area of a triangle is one half the product of the lengths of two sides and the sine of the included angle. Quick Check The area of the park is approximately 27,200 ft2. 10-5

  12. Trigonometry and Area Lesson 10-5 Lesson Quiz Find the area of each figure. Give answers to the nearest unit. 1. regular hexagon with perimeter 90 ft 2. regular pentagon with radius 12 m 3. regular polygon with 12 sides of length 1 in. 585 ft2 342 m2 11 in2 4. 5. 490 mm2 70 yd2 10-5

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