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6. 12,080 in. 2 7. 2475 in. 2 8. 1168.5 m 2 9. 2192.4 cm 2 10. 841.8 ft 2 11. 27.7 in. 2

Pages 382-385 Exercises. 1. m 1 = 120; m 2 = 60; m 3 = 30 2. m 4 = 90; m 5 = 45; m 6 = 45 3. m 7 = 60; m 8 = 30; m 9 = 60 4. 2144.475 cm 2 5. 2851.8 ft 2. 14. 384 3 in. 2 15. 300 3 ft 2 16. 162 3 m 2 17. 75 3 m 2 18. 12 3 in. 2

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6. 12,080 in. 2 7. 2475 in. 2 8. 1168.5 m 2 9. 2192.4 cm 2 10. 841.8 ft 2 11. 27.7 in. 2

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  1. Pages 382-385 Exercises 1.m 1 = 120; m 2 = 60; m 3 = 30 2.m 4 = 90; m 5 = 45; m 6 = 45 3.m 7 = 60; m 8 = 30; m 9 = 60 4. 2144.475 cm2 5. 2851.8 ft2 14. 384 3 in.2 15. 300 3 ft2 16. 162 3 m2 17. 75 3 m2 18. 12 3 in.2 19. a. 72 b. 54 20. a. 45 b. 67.5 GEOMETRY LESSON 7-5 6. 12,080 in.2 7. 2475 in.2 8. 1168.5 m2 9. 2192.4 cm2 10. 841.8 ft2 11. 27.7 in.2 12. 93.5 m2 13. 72 cm2

  2. 24. (continued) d. Answers may vary. Sample: About 4 in.; the length of a side of a pentagon should be between 3.7 in. and 6 in. 25.m 1 = 36; m 2 = 18; m 3 = 72 26. The apothem is one leg of a rt. and the radius is the hypotenuse. 27. 73 cm2 28. 130 in.2 29. 27 m2 30. 103 ft2 31. 220 cm2 32. a–c. regular octagon GEOMETRY LESSON 7-5 21. a. 40 b. 70 22. a. 30 b. 75 23. 310.4 ft2 24. a. 9.1 in. b. 6 in. c. 3.7 in.

  3. 32. (continued) d. Construct a 60° angle with vertex at circle’s center. 33. 600 3 m2 34. Check students’ work. 35. 128 cm2 36. 24 3 cm2, 41.6 cm2 37. 900 3 m2, 1558.8 m2 38. 100 ft2 39. 16 3 in.2, 27.7 in.2 40. m2, 70.1 m2 41. a.b = s; h = s A = bh A = s • s A = s2 3 41. (continued) b. apothem = ; A = ap = (3s) A = s2 3 1 2 1 2 1 2 81 3 2 s 3 6 s 3 6 1 2 1 2 3 2 3 2 GEOMETRY LESSON 7-5

  4. 7.6 Circles and Arcs Objective: To find the measures of central angles and arcs To find the circumference and arc length

  5. Definitions • Circle – the set of all points equidistant from a given point called the center • CenterofaCircle – the point from which all points are equidistant • Radius – a segment that has one endpoint at the center and the other endpoint on the circle

  6. Definitions • CongruentCircles – circles that have congruent radii • Diameter – a segment that contains the center of a circle and has both endpoints on the circle • CentralAngle – an angle whose vertex is the center of the circle

  7. Definitions • Circumference – the distance around the circle • Pi (∏) – the ration of the circumference of a circle to its diameter

  8. Examples • What if we are given a pie chart that represents data that have been collected? • How can we find the measure of the arc or the measure of the angle?

  9. A researcher surveyed 2000 members of a club to find their ages. The graph shows the survey results. Find the measure of each central angle in the circle graph. Circles and Arcs GEOMETRY LESSON 7-6 Because there are 360° in a circle, multiply each percent by 360 to find the measure of each central angle. 65+ : 25% of 360 = 0.25 • 360 = 90 45–64: 40% of 260 = 0.4 • 360 = 144 25–44: 27% of 360 = 0.27 • 360 = 97.2 Under 25: 8% of 360 = 0.08 • 360 = 28.8

  10. 132 1320 Circles and Arcs • Some info to really help • The measure of the arc is the same as the measure of the central angle which creates that arc

  11. R R T S P T S P R T S P Arcs A minor arc is smaller than a semicircle A major arc is greater than a semicircle A semicircle is half of a circle

  12. . Identify the minor arcs, major arcs, and semicircles in P with point A as an endpoint. Minor arcs are smaller than semicircles. Two minor arcs in the diagram have point A as an endpoint, AD and AE. Major arcs are larger than semicircles. Two major arcs in the diagram have point A as an endpoint, ADE and AED. Two semicircles in the diagram have point A as an endpoint, ADB and AEB. Circles and Arcs GEOMETRY LESSON 7-6

  13. Arcs • AdjacentArcs – arcs of the same circle that have exactly one point in common • CongruentArcs – arcs that have the same measure and are in the same circle or in congruent circles • ConcentricCircles – circles that lie in the same plane and have the same center • Arclength – a fraction of a circle’s circumference

  14. Postulate 7-1: Arc Addition Postulate • The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. • mABC = mAB + mBC

  15. . Find mXY and mDXM in C. mXY = mXD + mDYArc Addition Postulate mXY = m XCD + mDYThe measure of a minor arc is the measure of its corresponding central angle. mXY = 56 + 40 Substitute. mXY = 96 Simplify. mDXM = mDX + mXWMArc Addition Postulate mDXM = 56 + 180 Substitute. mDXM = 236 Simplify. Circles and Arcs GEOMETRY LESSON 7-6

  16. Circumference • The circumference of a circle is the product of pi (∏) and the diameter. • C = ∏d or C = 2∏r

  17. A circular swimming pool with a 16-ft diameter will be enclosed in a circular fence 4 ft from the pool. What length of fencing material is needed? Round your answer to the next whole number. Draw a diagram of the situation. C = dFormula for the circumference of a circle C = (24) Substitute. C 3.14(24) Use 3.14 to approximate . C 75.36 Simplify. Circles and Arcs GEOMETRY LESSON 7-6 The pool and the fence are concentric circles. The diameter of the pool is 16 ft, so the diameter of the fence is 16 + 4 + 4 = 24 ft. Use the formula for the circumference of a circle to find the length of fencing material needed. About 76 ft of fencing material is needed. 7-6

  18. Arc Length • The length of an arc of a circle is the product of the ratio measure of the arc 360 and the circumference of the circle • Length of AB = mAB/360 *2∏r

  19. . Find the length of ADB in M in terms of . Because mAB = 150, mADB = 360 – 150 = 210. Arc Addition Postulate 210 360 length of ADB = • 2 (18) Substitute. mADB 360 length of ADB = • 2 rArc Length Formula The length of ADB is 21 cm. length of ADB = 21 Circles and Arcs GEOMETRY LESSON 7-6 7-6

  20. 1. A circle graph has a section marked “Potatoes: 28%.” What is the measure of the central angle of this section? 2. Explain how a major arc differs from a minor arc. Use O for Exercises 3–6. 3. Find mYW. 4. Find mWXS. 5. Suppose that P has a diameter 2 in. greater than the diameter of O. How much greater is its circumference? Leave your answer in terms of . 6. Find the length of XY. Leave your answer in terms of . . . . 2 9 Circles and Arcs GEOMETRY LESSON 7-6 100.8 A major arc is greater than a semicircle. A minor arc is smaller than a semicircle. 30 270

  21. Assignment • P. 389-390 • #1-32 odd, 34-39

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