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3 APPLY ASSIGNMENT GUIDE GUIDED PRACTICE Vocabulary Check ✓ A 1. Identify the center of polygon ABCDE. J BASIC B 5.88 2. Identify the radius of the polygon. 5 Day 1: pp. 672–673 Exs. 9–29 Day 2: pp. 673–675 Exs. 30–44, 48–52, 54–64 K 3. Identify a central angle of the polygon. ™BJC Concept Check ✓ Skill Check ✓ AVERAGE Day 1: pp. 672–673 Exs. 9–29 Day 2: pp. 673–675 Exs. 30–52, 54–64 In Exercises 1–4, use the diagram shown. ADVANCED 4. Identify a segment whose length is the apothem. J 4.05 E 5 Æ JK C 5. In a regular polygon, how do you find the D measure of each central angle? Divide 360° by the number of sides. 3 9 苶 4 6. What is the area of an equilateral triangle with 3 inch sides? }兹} ≈ 3.9 in.2 STOP SIGN The stop sign shown is a regular octagon. Its perimeter is about 80 inches and its height is about 24 inches. Day 1: pp. 672–673 Exs. 9–29 Day 2: pp. 673–675 Exs. 30–64 BLOCK SCHEDULE 7. What is the measure of each central angle? pp. 672–675 Exs. 9–52, 54–64 45° 8. Find the apothem, radius, and area of the stop sign. about 12 in., about 13 in., about 480 in.2 EXERCISE LEVELS Level A: Easier 9–15 Level B: More Difficult 16–52 Level C: Most Difficult 53 PRACTICE AND APPLICATIONS FINDING AREA Find the area of the triangle. STUDENT HELP Extra Practice to help you master skills is on p. 823. 9. 10. 5 HOMEWORK CHECK To quickly check student understanding of key concepts, go over the following exercises: Exs. 10, 18, 20, 22, 26, 28, 32, 38, 44. See also the Daily Homework Quiz: • Blackline Master (Chapter 11 Resource Book, p. 40) • Transparency (p. 81) 11. 5 7兹5 11 5 25兹3苶 }} ≈ 10.8 sq. units 4 121兹3苶 }} ≈ 52.4 sq. units 4 7兹5 245兹3苶 }} ≈ 106.1 sq. units 4 MEASURES OF CENTRAL ANGLES Find the measure of a central angle of a regular polygon with the given number of sides. 12. 9 sides 13. 12 sides 40° 14. 15 sides 30° 15. 180 sides 24° 2° FINDING AREA Find the area of the inscribed regular polygon shown. 16. A B 17. STUDENT HELP D D C C Example 1: Exs. 9–11, 17, 19, 25, 33 Example 2: Exs. 9–11, 17, 19, 25, 33 Example 3: Exs. 12–24, 26, 34 Example 4: Exs. 34, 45–49 G 20 E C 10兹3 B E D 108兹3苶 ≈ 187.1 sq. units 600兹3苶 ≈ 1039.2 sq. units PERIMETER AND AREA Find the perimeter and area of the regular polygon. 19. 20. 21. 10 30兹3苶 ≈ 52.0 units; 75兹3苶 ≈ 129.9 sq. units 672 B F 12 128 sq. units HOMEWORK HELP A 6 4兹2 F 18. A E 8 672 7兹5 Chapter 11 Area of Polygons and Circles 4 16兹2苶 units; 32 sq. units 15 150 tan 36° ≈ 109.0 units; 1125 tan 36° ≈ 817.36 sq. units PERIMETER AND AREA In Exercises 22–24, find the perimeter and area of the regular polygon. 147 3苶 兹 22. 42 units; }} ≈ 22. 23. 24. 2 127. 31 sq. units 23. 176 sin 22.5° ≈ 67.35 units; 9 968(sin 22.5°)(cos 22.5°) ≈ 11 342.24 sq. units 7 24. 216 sin 15° ≈ 55.9 units; 972(sin 15°)(cos 15°) = 243 sq. units 25. AREA Find the area of an equilateral triangle that has a height of 15 inches. 75兹3苶 ≈ 129.9 in.2 27. True; let † be the measure 26. A REA Find the area of a regular dodecagon (or 12-gon) that has 4 inch sides. of a central angle, n the 48(tan 75°) ≈ 179.14 in.2 number of sides, r the LOGICAL REASONING Decide whether the statement is true or false. radius, and P the perimeter. Explain your choice. As n grows bigger † will become smaller, so the 27. The area of a regular polygon of fixed radius r increases as the number of apothem, which is given by sides increases. See margin. † by r cos }} will get larger. 2 28. The apothem of a regular polygon is always less than the radius. See margin. The perimeter of the polygon, which is given 29. The radius of a regular polygon is always less than the side length. † False; for example, the radius of a regular hexagon is equal to the side length. by n 2rsin }} will grow 2 larger, too. Although the AREA In Exercises 30–32, find the area of the regular polygon. The area of factor involving the sine the portion shaded in red is given. Round answers to the nearest tenth. will get smaller, the 30. Area = 16兹3 苶 31. Area = 4 tan 67.5° 32. Area = tan 54° increase in n more than makes up for it. Consequently, the area, 1 q which is given by }}aP q 2 will increase. q 28. True; the apothem is a leg length in a right ¤ of 32 tan 67.5° ≈ 77.3 5 tan 54° ≈ 6.9 96兹3苶 ≈ 166.3 which the radius is the 33. USING THE AREA FORMULAS Show that the area of a regular hexagon is length of the hypotenuse. 冉 冊 six times the area of an equilateral triangle with the same side length. 冉 冉 TEACHING TIPS EXERCISES 16–21 Students may come up with slightly different answers to these problems. To get the most accurate answer possible, have students carry trigonometric expressions or radical expressions through until the very last step, then round, as is shown in Example 3 on page 671. COMMON ERROR EXERCISES 22–24 Students might use the central angle when using trigonometry to find the apothem and side length. Point out that the acute angle in the triangle formed by a radius and a segment whose length is the apothem is half the measure of the central angle of the polygon. Also, students might forget to double the length of the leg of this triangle to get the side length. This error would result in an area that is half the actual area of the polygon. 冊冊 33. Let s = the length of a side 1 1 of the hexagon and of the Hint: Show that for a hexagon with side lengths s, ᎏᎏ aP = 6 • ᎏᎏ兹3苶s 2 . 2 4 equilateral triangle. The apothem of the hexagon 34. BASALTIC COLUMNS Suppose the top of one of the columns along the 1 is }}兹3苶s and the perimeter Giant’s Causeway (see p. 659) is in the shape of a regular hexagon with a 2 of the hexagon is 6s. The diameter of 18 inches. What is its apothem? 4.5兹3苶 ≈ 7.8 in. area of the hexagon, then, is 1 1 1 A = }}aP = }} }}兹3苶s • 6s, CONSTRUCTION In Exercises 35–39, use a straightedge and a 2 2 2 3 or }}兹3苶s2. The area of an compass to construct a regular hexagon and an equilateral triangle. 2 35–37. Check drawings. Æ equilateral triangle with 35. Draw AB with a length of 1 inch. Open the side length s is compass to 1 inch and draw a circle with that radius. A B 1 A = ᎏᎏ兹3苶s2. Six of these 4 36. Using the same compass setting, mark off equal equilateral triangles parts along the circle. together (forming the hexagon), then, would have 37. Connect the six points where the compass marks s2 3苶 3s2兹3苶 and circle intersect to draw a regular hexagon. area 6 • }兹} = }} . 冉 4 冊 2 The two results are the same. 3 3苶 兹 38. What is the area of the hexagon? }2} ≈ 2.60 in.2 39. Writing Explain how you could use this construction to construct an equilateral triangle. Draw segments connecting 3 of the compass marks, skipping every second mark. 11.2 Areas of Regular Polygons 673 673 STUDENT HELP INT STUDENT HELP NOTES NE ER T HOMEWORK HELP Visit our Web site www.mcdougallittell for help with construction in Exs. 40–44. Homework Help Students can find help for Exs. 40–44 at www.mcdougallittell.com. The information can be printed out for students who don’t have access to the Internet. CONSTRUCTION In Exercises 40–44, use a straightedge and a compass to construct a regular pentagon as shown in the diagrams below. 40–44. Check drawings. A CAREER NOTE EXERCISES 45 AND 46 Additional information about astronomers is available at www.mcdougallittell.com. q B A Exs. 40, 41 q B q A Ex. 42 Exs. 43, 44 Æ 40. Draw a circle with center Q. Draw a diameter AB. Construct the perpendicular Æ bisector of AB and label its intersection with the circle as point C. ENGLISH LEARNERS EXERCISES 45–46 Understanding terminology in word problems can be difficult for English learners. For Exercises 45–46, students may have difficulty understanding what a primary mirror is, and thus have trouble interpeting the photograph. Æ 41. Construct point D, the midpoint of QB. 42. Place the compass point at D. Open the compass to the length DC and draw Æ Æ an arc from C so it intersects AB at a point, E. Draw CE. 43. Open the compass to the length CE. Starting at C, mark off equal parts along the circle. 44. Connect the five points where the compass marks and circle intersect to draw a regular pentagon. What is the area of your pentagon? FOCUS ON CAREERS Check results. TELESCOPES In Exercises 45 and 46, use the following information. The Hobby-Eberly Telescope in Fort Davis, Texas, is the largest optical telescope in North America. The primary mirror for the telescope consists of 91 smaller mirrors forming a hexagon shape. Each of the smaller mirror parts is itself a hexagon with side length 0.5 meter. 45. What is the apothem of one of the smaller mirrors? }1}兹3苶 ≈ 0.43 m 4 46. Find the perimeter and area of one of the smaller mirrors. 3.0 m, about 0.65 m 2 RE • Practice Levels A, B, and C (Chapter 11 Resource Book, p. 30) • Reteaching with Practice (Chapter 11 Resource Book, p. 33) • See Lesson 11.2 of the Personal Student Tutor For more Mixed Review: • Search the Test and Practice Generator for key words or specific lessons. 674 ASTRONOMERS use physics and mathematics to study the universe, including the sun, moon, planets, stars, and galaxies. INT For Lesson 11.2: FE L AL I ADDITIONAL PRACTICE AND RETEACHING B NE ER T CAREER LINK www.mcdougallittell.com TILING In Exercises 47–49, use the following information. You are tiling a bathroom floor with tiles that are regular hexagons, as shown. Each tile has 6 inch sides. You want to choose different colors so that no two adjacent tiles are the same color. 47. What is the minimum number of colors that you can use? 3 colors 48. What is the area of each tile? 54兹3苶 ≈ 93.5 in.2 49. The floor that you are tiling is rectangular. Its width is 6 feet and its length is 8 feet. At least how many tiles of each color will you need? about 25 tiles 674 Chapter 11 Area of Polygons and Circles Test Preparation QUANTITATIVE COMPARISON In Exercises 50–52, choose the statement that is true about the given quantities. ¡ B ¡ C ¡ D ¡ A DAILY HOMEWORK QUIZ The quantity in column B is greater. The relationship cannot be determined from the given information. Column B r B 1 1 s P N 1 9 9 81兹3苶 } ≈ 35.1 square units 4 q 1 50. m™APB m™MQN B 51. Apothem r Apothem s A 52. Perimeter of octagon with center P Perimeter of heptagon with center Q A A A B 14 5 of ABCDE by using two methods. First, use the 1 2 2. Find the measure of the central angle of a regular polygon with 10 sides. 36° 3. Find the area of the inscribed regular polygon shown. P 53. USING DIFFERENT METHODS Find the area 1 2 B E formula A = ᎏᎏaP, or A = ᎏᎏ a • ns. Second, add www.mcdougallittell.com 9 M A EXTRA CHALLENGE Transparency Available 1. Find the area of the triangle. The two quantities are equal. Column A ★ Challenge 4 ASSESS The quantity in column A is greater. C P the areas of the smaller polygons. Check that both methods yield the same area. E D 392 square units 4. Find the area of a regular hexagon that has 4-inch sides. C D 1 1 1 1 }}aP ≈ }}(3.44)(25) ≈ 43; }}a • ns ≈ }}(3.44)(5)(5) ≈ 43; 2 2 2 2 area of †ABE + area of EDCB ≈ 11.9 + 31.1 ≈ 43 7 2 24兹3苶 ≈ 41.6 MIXED REVIEW EXTRA CHALLENGE NOTE SOLVING PROPORTIONS Solve the proportion. (Review 8.1 for 11.3) x 11 11 54. ᎏᎏ = ᎏᎏ }2} 6 12 12 13 56. ᎏᎏ = ᎏᎏ º91 x+7 x 20 15 55. ᎏᎏ = ᎏᎏ 3 4 x USING SIMILAR POLYGONS In the diagram shown, ¤ABC ~ ¤DEF. Use the figures to determine whether the statement is true. (Review 8.3 for 11.3) AC DF DF EF + DE + DF 58. ᎏᎏ = ᎏᎏ true 59. ᎏ = ᎏᎏ BC EF AC BC + AB + AC Æ Æ º33 x x+6 57. ᎏᎏ = ᎏᎏ 11 9 A D 4 3 true B 60. ™B £ ™E true 61. BC £ EF false E C F FINDING SEGMENT LENGTHS Find the value of x. (Review 10.5) 62. 6 63. 7 x 7 9 x 12 14 8 10 64. 12 8 4 x 11.2 Areas of Regular Polygons 675 Challenge problems for Lesson 11.2 are available in blackline format in the Chapter 11 Resource Book, p. 37 and at www.mcdougallittell.com. ADDITIONAL TEST PREPARATION 1. OPEN ENDED Draw a regular polygon and show how to find its area. Check student’s work. 2. WRITING Explain why you might use the Pythagorean Theorem when finding the area of a regular polygon. Sample answer: If you know the radius and side length of the regular polygon, you can use the Pythagorean Theorem with the radius and half the side length to find the apothem. 675