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Circles Theorems: Real-World Examples & Chord Equidistance

Explore arc and chord relationships in circles with real-world examples, congruency theorems, and chord equidistance concepts. Solve problems using arc measures and chord lengths.

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Circles Theorems: Real-World Examples & Chord Equidistance

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  1. LESSON 10–3 Arcs and Chords

  2. Five-Minute Check (over Lesson 10–2) TEKS 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. 35 B. 55 C. 66 D. 72 5-Minute Check 2

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

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

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

  8. 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

  9. Targeted TEKS G.12(A) Apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non-contextual problems. Mathematical Processes G.1(B), G.1(G) TEKS

  10. 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

  11. Concept

  12. 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

  13. Use Congruent Chords to Find Arc Measure Example 1

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

  15. Use Congruent Arcs to Find Chord Lengths Example 2

  16. 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

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

  18. Concept

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

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

  21. 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

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

  23. 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

  24. 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

  25. 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

  26. Concept

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

  28. 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

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

  30. LESSON 10–3 Arcs and Chords

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