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Geometry Section 10-3 1112

J
Jimbo Lamb

This document covers theorems and examples relating to arcs and chords in circles. It states four theorems: 1) Congruent minor arcs have congruent chords, 2) A diameter or radius perpendicular to a chord bisects the chord and arc, 3) The perpendicular bisector of a chord is a diameter or radius, and 4) Congruent chords are equidistant from the center. It then provides five examples applying these theorems to find measures of arcs, chords, and radii given various circle properties. The document concludes with sample problems for students to practice these concepts.

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Section 10-3 Arcs and Chords Monday, May 14, 2012
Essential Questions • How do you recognize and use relationships between arcs and chords? • How do you recognize and use relationships between arcs, chords, and diameters? Monday, May 14, 2012
Theorems 10.2 - Congruent Minor Arcs: 10.3 - Perpendicularity: 10.4 - Perpendicularity: 10.5 - Congruent Chords: Monday, May 14, 2012
Theorems 10.2 - Congruent Minor Arcs: In the same or congruent circles, two minor arcs are congruent IFF their corresponding chords are congruent 10.3 - Perpendicularity: 10.4 - Perpendicularity: 10.5 - Congruent Chords: Monday, May 14, 2012
Theorems 10.2 - Congruent Minor Arcs: In the same or congruent circles, two minor arcs are congruent IFF their corresponding chords are congruent 10.3 - Perpendicularity: If a diameter or radius of a circle is perpendicular to a chord, then it bisects the chord and its arc 10.4 - Perpendicularity: 10.5 - Congruent Chords: Monday, May 14, 2012
Theorems 10.2 - Congruent Minor Arcs: In the same or congruent circles, two minor arcs are congruent IFF their corresponding chords are congruent 10.3 - Perpendicularity: If a diameter or radius of a circle is perpendicular to a chord, then it bisects the chord and its arc 10.4 - Perpendicularity: The perpendicular bisector of a chord is a diameter or radius of the circle 10.5 - Congruent Chords: Monday, May 14, 2012
Theorems 10.2 - Congruent Minor Arcs: In the same or congruent circles, two minor arcs are congruent IFF their corresponding chords are congruent 10.3 - Perpendicularity: If a diameter or radius of a circle is perpendicular to a chord, then it bisects the chord and its arc 10.4 - Perpendicularity: The perpendicular bisector of a chord is a diameter or radius of the circle 10.5 - Congruent Chords: In the same or congruent circles, two chords are congruent IFF they are equidistant from the center Monday, May 14, 2012
Example 1  = 90°. Find mAB. In  X, AB ≅ CD and mCD  Monday, May 14, 2012
Example 1  = 90°. Find mAB. In  X, AB ≅ CD and mCD   = 90° mAB Monday, May 14, 2012
Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  Monday, May 14, 2012
Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 Monday, May 14, 2012
Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 2x = 8 Monday, May 14, 2012
Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 2x = 8 x=4 Monday, May 14, 2012
Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 2x = 8 x=4 WX = 7(4) − 2 Monday, May 14, 2012
Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 2x = 8 x=4 WX = 7(4) − 2 WX = 28 − 2 Monday, May 14, 2012
Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 2x = 8 x=4 WX = 7(4) − 2 WX = 28 − 2 WX = 26 Monday, May 14, 2012
Example 3  =150°. Find mDE. In  G, mDEF  Monday, May 14, 2012
Example 3  =150°. Find mDE. In  G, mDEF   = 1 mDEF mDE  2 Monday, May 14, 2012
Example 3  =150°. Find mDE. In  G, mDEF   = 1 mDEF mDE  2  = 1 (150) mDE 2 Monday, May 14, 2012
Example 3  =150°. Find mDE. In  G, mDEF   = 1 mDEF mDE  2  = 1 (150) mDE 2  = 75° mDE Monday, May 14, 2012
Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. Monday, May 14, 2012
Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. Monday, May 14, 2012
Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 Monday, May 14, 2012
Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 4 +b =9 2 2 2 Monday, May 14, 2012
Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 4 +b =9 2 2 2 16 + b = 81 2 Monday, May 14, 2012
Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 4 +b =9 2 2 2 16 + b = 81 2 b = 65 2 Monday, May 14, 2012
Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 4 +b =9 2 2 2 16 + b = 81 2 b = 65 2 b = 65 Monday, May 14, 2012
Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 4 +b =9 2 2 2 16 + b = 81 2 b = 65 2 b = 65 inches or ≈ 8.06 inches Monday, May 14, 2012
Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. Monday, May 14, 2012
Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 Monday, May 14, 2012
Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 2x = 6 Monday, May 14, 2012
Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 2x = 6 x=3 Monday, May 14, 2012
Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 2x = 6 x=3 PQ = 4(3) − 3 Monday, May 14, 2012
Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 2x = 6 x=3 PQ = 4(3) − 3 PQ = 12 − 3 Monday, May 14, 2012
Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 2x = 6 x=3 PQ = 4(3) − 3 PQ = 12 − 3 PQ = 9 Monday, May 14, 2012
Check Your Understanding p. 704 #1 - 6 Monday, May 14, 2012
Problem Set Monday, May 14, 2012
Problem Set p. 705 #7-33 odd, 45, 49, 51 "I may not have gone where I intended to go, but I think I have ended up where I needed to be." - Douglas Adams Monday, May 14, 2012

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Geometry Section 10-3 1112

  • 1. Section 10-3 Arcs and Chords Monday, May 14, 2012
  • 2. Essential Questions • How do you recognize and use relationships between arcs and chords? • How do you recognize and use relationships between arcs, chords, and diameters? Monday, May 14, 2012
  • 3. Theorems 10.2 - Congruent Minor Arcs: 10.3 - Perpendicularity: 10.4 - Perpendicularity: 10.5 - Congruent Chords: Monday, May 14, 2012
  • 4. Theorems 10.2 - Congruent Minor Arcs: In the same or congruent circles, two minor arcs are congruent IFF their corresponding chords are congruent 10.3 - Perpendicularity: 10.4 - Perpendicularity: 10.5 - Congruent Chords: Monday, May 14, 2012
  • 5. Theorems 10.2 - Congruent Minor Arcs: In the same or congruent circles, two minor arcs are congruent IFF their corresponding chords are congruent 10.3 - Perpendicularity: If a diameter or radius of a circle is perpendicular to a chord, then it bisects the chord and its arc 10.4 - Perpendicularity: 10.5 - Congruent Chords: Monday, May 14, 2012
  • 6. Theorems 10.2 - Congruent Minor Arcs: In the same or congruent circles, two minor arcs are congruent IFF their corresponding chords are congruent 10.3 - Perpendicularity: If a diameter or radius of a circle is perpendicular to a chord, then it bisects the chord and its arc 10.4 - Perpendicularity: The perpendicular bisector of a chord is a diameter or radius of the circle 10.5 - Congruent Chords: Monday, May 14, 2012
  • 7. Theorems 10.2 - Congruent Minor Arcs: In the same or congruent circles, two minor arcs are congruent IFF their corresponding chords are congruent 10.3 - Perpendicularity: If a diameter or radius of a circle is perpendicular to a chord, then it bisects the chord and its arc 10.4 - Perpendicularity: The perpendicular bisector of a chord is a diameter or radius of the circle 10.5 - Congruent Chords: In the same or congruent circles, two chords are congruent IFF they are equidistant from the center Monday, May 14, 2012
  • 8. Example 1  = 90°. Find mAB. In  X, AB ≅ CD and mCD  Monday, May 14, 2012
  • 9. Example 1  = 90°. Find mAB. In  X, AB ≅ CD and mCD   = 90° mAB Monday, May 14, 2012
  • 10. Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  Monday, May 14, 2012
  • 11. Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 Monday, May 14, 2012
  • 12. Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 2x = 8 Monday, May 14, 2012
  • 13. Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 2x = 8 x=4 Monday, May 14, 2012
  • 14. Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 2x = 8 x=4 WX = 7(4) − 2 Monday, May 14, 2012
  • 15. Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 2x = 8 x=4 WX = 7(4) − 2 WX = 28 − 2 Monday, May 14, 2012
  • 16. Example 2  ≅ YZ . Find WX. In the figure,  A ≅ B and WX  7x − 2 = 5x + 6 2x = 8 x=4 WX = 7(4) − 2 WX = 28 − 2 WX = 26 Monday, May 14, 2012
  • 17. Example 3  =150°. Find mDE. In  G, mDEF  Monday, May 14, 2012
  • 18. Example 3  =150°. Find mDE. In  G, mDEF   = 1 mDEF mDE  2 Monday, May 14, 2012
  • 19. Example 3  =150°. Find mDE. In  G, mDEF   = 1 mDEF mDE  2  = 1 (150) mDE 2 Monday, May 14, 2012
  • 20. Example 3  =150°. Find mDE. In  G, mDEF   = 1 mDEF mDE  2  = 1 (150) mDE 2  = 75° mDE Monday, May 14, 2012
  • 21. Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. Monday, May 14, 2012
  • 22. Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. Monday, May 14, 2012
  • 23. Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 Monday, May 14, 2012
  • 24. Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 4 +b =9 2 2 2 Monday, May 14, 2012
  • 25. Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 4 +b =9 2 2 2 16 + b = 81 2 Monday, May 14, 2012
  • 26. Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 4 +b =9 2 2 2 16 + b = 81 2 b = 65 2 Monday, May 14, 2012
  • 27. Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 4 +b =9 2 2 2 16 + b = 81 2 b = 65 2 b = 65 Monday, May 14, 2012
  • 28. Example 4 In  C, AB =18 inches and EF = 8 inches. Find CD. CF is a radius. a +b =c 2 2 2 4 +b =9 2 2 2 16 + b = 81 2 b = 65 2 b = 65 inches or ≈ 8.06 inches Monday, May 14, 2012
  • 29. Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. Monday, May 14, 2012
  • 30. Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 Monday, May 14, 2012
  • 31. Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 2x = 6 Monday, May 14, 2012
  • 32. Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 2x = 6 x=3 Monday, May 14, 2012
  • 33. Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 2x = 6 x=3 PQ = 4(3) − 3 Monday, May 14, 2012
  • 34. Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 2x = 6 x=3 PQ = 4(3) − 3 PQ = 12 − 3 Monday, May 14, 2012
  • 35. Example 5 In  P, EF = GH = 24, PQ = 4x − 3, and PR = 2x + 3. Find PQ. 4x − 3 = 2x + 3 2x = 6 x=3 PQ = 4(3) − 3 PQ = 12 − 3 PQ = 9 Monday, May 14, 2012
  • 36. Check Your Understanding p. 704 #1 - 6 Monday, May 14, 2012
  • 38. Problem Set p. 705 #7-33 odd, 45, 49, 51 "I may not have gone where I intended to go, but I think I have ended up where I needed to be." - Douglas Adams Monday, May 14, 2012