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12.3 Arcs and Chords

Learn about arc and chord properties in circles, find arc measures, and apply theorems for congruent chords. Check out examples and theorems to deepen your understanding. Quiz coming up—be prepared!

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12.3 Arcs and Chords

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  1. 12.3 Arcs and Chords Geometry

  2. Objectives/Assignment • Use properties of arcs of circles, as applied. • Use properties of chords of circles. • Reminder Quiz Tomorrow!!!!!!

  3. Ex. 1: Finding Measures of Arcs • Find the measure of each arc of R. 80°

  4. Ex. 1: Finding Measures of Arcs • Find the measure of each arc of R. Solution: is a minor arc, so m = mMRN = 80° 80°

  5. Ex. 1: Finding Measures of Arcs • Find the measure of each arc of R. Solution: is a major arc, so m = 360° – 80° = 280° 80°

  6. Ex. 1: Finding Measures of Arcs • Find the measure of each arc of R. Solution: is a semicircle, so m = 180° 80°

  7. Ex. 2: Finding Measures of Arcs • Find the measure of each arc. m = m + m = 40° + 80° = 120° 40° 80° 110°

  8. Ex. 2: Finding Measures of Arcs • Find the measure of each arc. m = m + m = 120° + 110° = 230° 40° 80° 110°

  9. Ex. 2: Finding Measures of Arcs • Find the measure of each arc. m = 360° - m = 360° - 230° = 130° 40° 80° 110°

  10. if and only if  Theorem 12.6 • In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

  11. If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.  ,  Theorem 12.7

  12. If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. Converse of Theorem 12.7 is a diameter of the circle.

  13. Because AD  DC, and  . So, m = m Ex. 4: Using Theorem 12.6 (x + 40)° • You can use Theorem 10.4 to find m . 2x° 2x = x + 40 Substitute Subtract x from each side. x = 40

  14. Theorem 12.7 can be used to locate a circle’s center as shown in the next few slides. Step 1: Draw any two chords that are not parallel to each other. Finding the Center of a Circle

  15. Step 2: Draw the perpendicular bisector of each chord. These are the diameters. Finding the Center of a Circle

  16. Step 3: The perpendicular bisectors intersect at the circle’s center. Finding the Center of a Circle

  17. In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. AB  CD if and only if EF  EG. Theorem 12.9

  18. AB = 8; DE = 8, and CD = 5. Find CF. Ex. 7: Using Theorem 12.9

  19. Ex. 7: Using Theorem 12.9 Because AB and DE are congruent chords, they are equidistant from the center. So CF  CG. To find CG, first find DG. CG  DE, so CG bisects DE. Because DE = 8, DG = =4.

  20. Ex. 7: Using Theorem 12.9 Then use DG to find CG. DG = 4 and CD = 5, so ∆CGD is a 3-4-5 right triangle. So CG = 3. Finally, use CG to find CF. Because CF  CG, CF = CG = 3

  21. Reminders: • Quiz after 12.3 • Last day for seniors is this Friday, make sure you return your books!

  22. Homework: Finish the worksheet 12.3 Last day for seniors is this Friday, make sure you return your books!

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