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Geometry Lesson: Arc Measures, Chord Lengths, and Perpendicular Chords

This lesson covers the use of congruent chords to find arc measure, the use of congruent arcs to find chord lengths, and the use of a radius or diameter perpendicular to a chord. It also includes examples and real-world applications.

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Geometry Lesson: Arc Measures, Chord Lengths, and Perpendicular Chords

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 10–2) CCSS Then/Now Theorem 10.2 Example 1: Real-World Example: Use Congruent Chords to Find Arc Measure Example 2: Use Congruent Arcs to Find Chord Lengths Theorems Example 3: Use a Radius Perpendicular to a Chord Example 4: Real-World Example: Use a Diameter Perpendicular to a Chord Theorem 10.5 Example 5: Chords Equidistant from Center Lesson Menu

  3. A. 105 B. 114 C. 118 D. 124 5-Minute Check 1

  4. A. 105 B. 114 C. 118 D. 124 5-Minute Check 1

  5. A. 35 B. 55 C. 66 D. 72 5-Minute Check 2

  6. A. 35 B. 55 C. 66 D. 72 5-Minute Check 2

  7. A. 125 B. 130 C. 135 D. 140 5-Minute Check 3

  8. A. 125 B. 130 C. 135 D. 140 5-Minute Check 3

  9. A. 160 B. 150 C. 140 D. 130 5-Minute Check 4

  10. A. 160 B. 150 C. 140 D. 130 5-Minute Check 4

  11. A. 180 B. 190 C. 200 D. 210 5-Minute Check 5

  12. A. 180 B. 190 C. 200 D. 210 5-Minute Check 5

  13. Dianne runs around a circular track that has a radius of 55 feet. After running three quarters of the distance around the track, how far has she run? A. 129.6 ft B. 165 ft C. 259.2 ft D. 345.6 ft 5-Minute Check 6

  14. Dianne runs around a circular track that has a radius of 55 feet. After running three quarters of the distance around the track, how far has she run? A. 129.6 ft B. 165 ft C. 259.2 ft D. 345.6 ft 5-Minute Check 6

  15. Content Standards G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 4 Model with mathematics. 3 Construct viable arguments and critique the reasoning of others. CCSS

  16. You used the relationships between arcs and angles to find measures. • Recognize and use relationships between arcs and chords. • Recognize and use relationships between arcs, chords, and diameters. Then/Now

  17. Concept

  18. Jewelry A circular piece of jade is hung from a chain by two wires around the stone. JM  KL and = 90. Find . Use Congruent Chords to Find Arc Measure Example 1

  19. Use Congruent Chords to Find Arc Measure Example 1

  20. Use Congruent Chords to Find Arc Measure Example 1

  21. A. 42.5 B. 85 C. 127.5 D. 170 Example 1

  22. A. 42.5 B. 85 C. 127.5 D. 170 Example 1

  23. Use Congruent Arcs to Find Chord Lengths Example 2

  24. Use Congruent Arcs to Find Chord Lengths WX = YZ Definition of congruent segments 7x – 2 = 5x + 6 Substitution 7x = 5x + 8 Add 2 to each side. 2x = 8 Subtract 5x from each side. x = 4 Divide each side by 2. So, WX = 7x – 2 = 7(4) – 2 or 26. Answer: Example 2

  25. Use Congruent Arcs to Find Chord Lengths WX = YZ Definition of congruent segments 7x – 2 = 5x + 6 Substitution 7x = 5x + 8 Add 2 to each side. 2x = 8 Subtract 5x from each side. x = 4 Divide each side by 2. So, WX = 7x – 2 = 7(4) – 2 or 26. Answer:WX = 26 Example 2

  26. A. 6 B. 8 C. 9 D. 13 Example 2

  27. A. 6 B. 8 C. 9 D. 13 Example 2

  28. Concept

  29. Use a Radius Perpendicular to a Chord Answer: Example 3

  30. Answer: Use a Radius Perpendicular to a Chord Example 3

  31. A. 14 B. 80 C. 160 D. 180 Example 3

  32. A. 14 B. 80 C. 160 D. 180 Example 3

  33. CERAMIC TILE In the ceramic stepping stone below, diameter AB is 18 inches long and chord EF is 8 inches long. Find CD. Use a Diameter Perpendicular to a Chord Example 4

  34. Step 1 Draw radius CE. Use a Diameter Perpendicular to a Chord This forms right ΔCDE. Example 4

  35. Since diameter AB is perpendicular to EF, AB bisects chord EF by Theorem 10.3. So, DE = (8) or 4 inches. 1 __ 2 Use a Diameter Perpendicular to a Chord Step 2 Find CE and DE. Since AB = 18 inches, CB = 9 inches. All radii of a circle are congruent, soCE = 9 inches. Example 4

  36. Use a Diameter Perpendicular to a Chord Step 3 Use the Pythagorean Theorem to find CD. CD2 + DE2 = CE2 Pythagorean Theorem CD2 + 42 = 92 Substitution CD2 + 16 = 81 Simplify. CD2 = 65 Subtract 16 from each side. Take the positive square root. Answer: Example 4

  37. Answer: Use a Diameter Perpendicular to a Chord Step 3 Use the Pythagorean Theorem to find CD. CD2 + DE2 = CE2 Pythagorean Theorem CD2 + 42 = 92 Substitution CD2 + 16 = 81 Simplify. CD2 = 65 Subtract 16 from each side. Take the positive square root. Example 4

  38. In the circle below, diameter QS is 14 inches long and chord RT is 10 inches long. Find VU. A. 3.87 B. 4.25 C. 4.90 D. 5.32 Example 4

  39. In the circle below, diameter QS is 14 inches long and chord RT is 10 inches long. Find VU. A. 3.87 B. 4.25 C. 4.90 D. 5.32 Example 4

  40. Concept

  41. Since chords EF and GH are congruent, they are equidistant from P. So, PQ = PR. Chords Equidistant from Center Example 5

  42. Chords Equidistant from Center • PQ = PR • 4x – 3 = 2x + 3 Substitution • x = 3 Simplify. • So, PQ = 4(3) – 3 or 9 • Answer: Example 5

  43. Chords Equidistant from Center • PQ = PR • 4x – 3 = 2x + 3 Substitution • x = 3 Simplify. • So, PQ = 4(3) – 3 or 9 • Answer:PQ = 9 Example 5

  44. A. 7 B. 10 C. 13 D. 15 Example 5

  45. A. 7 B. 10 C. 13 D. 15 Example 5

  46. End of the Lesson

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