Lesson2 unit-10 (2)
- 1. Relationship Between Chords and Arcs A. Bisected Chord Theorem B. Apothem C. Congruent Chords Theorem
- 2. A. Bisected Chord Theorem If a radius of a circle bisects a chord on the same circle, then the radius and the chord are perpendicular. Example: In the figure to the right, the radius 𝐴𝐵 bisects the chord 𝐶𝐷 at 𝑋. It follows that 𝐴𝐵 ⊥ 𝐶𝐷.
- 3. A. Bisected Chord Theorem 1. point of bisection or midpoint X of the bisecting radius 𝑨𝑩 2. As a result, a. segment 𝐶𝑋 = 𝑋𝐷 b. arc 𝑪𝑩 = 𝑩𝑫 and c. radius 𝑨𝑩 ⊥ 𝑪𝑫
- 4. A. Bisected Chord Theorem 1. point of bisection or midpoint X of the bisecting radius 𝑨𝑩 2. segment 𝐶𝑋 and 𝑋𝐷 3. arc 𝑪𝑩 and 𝑩𝑫
- 5. Learn about It! 2 Apothem a distance of a chord to the center of the circle Example: In the figure to the right, 𝑉𝑇 represents the distance between 𝑈𝑊 and the center 𝑇. As such 𝑉𝑇 is the apothem. In the same manner, 𝑇𝑌 is also the apothem between 𝑋𝑍 and the center 𝑇.
- 7. Learn about It! 3 Congruent Chords Theorem If two chords of a circle are congruent, then their apothems and intercepted arcs are also congruent. Example: The figure to the right illustrates circle 𝑂 with congruent chords 𝐴𝐵 and 𝐷𝐸. It follows that the apothems 𝐹𝑂 and 𝐶𝑂 are congruent as well as the intercepted arcs 𝐴𝐵 and 𝐷𝐸.
- 8. C. Congruent Chords Theorem 1. chord 𝐴𝐵 and 𝐷𝐸 2. arc 𝐴𝐵 and 𝐷𝐸 if 𝐴𝐵 = 𝐷𝐸 also therefore, 𝐴𝐵 = 𝐷𝐸
- 9. Example 1: In a given circle 𝑂, the chords 𝐴𝐵 and 𝐵𝐶 are congruent. What can be concluded between 𝐴𝐵 and 𝐵𝐶?
- 10. Try It! Example 1: In a given circle 𝑂, the chords 𝐴𝐵 and 𝐵𝐶 are congruent. What can be concluded between 𝐴𝐵 and 𝐵𝐶? Solution: Let us illustrate the problem. Since 𝐴𝐵 ≅ 𝐵𝐶, it follows that 𝑨𝑩 ≅ 𝑩𝑪 according to the Congruent Chords Theorem.
- 11. Example 2: In circle 𝑂, the chords 𝐴𝐵 and 𝐶𝐷 are congruent. If 𝑚 𝐴𝐵 = 2𝑥 + 13 ° and 𝑚 𝐶𝐷 = 3𝑥
- 12. Example 2: In circle 𝑂, the chords 𝐴𝐵 and 𝐶𝐷 are congruent. If 𝑚 𝐴𝐵 = 2𝑥 + 13 ° and 𝑚 𝐶𝐷 = 3𝑥 − 2 ° , find 𝑚 𝐴𝐵 and 𝑚 𝐶𝐷. Solution 2.1: Since 𝑚 𝐴𝐵 = 𝑚 𝐶𝐷, we can solve for the value of 𝑥 by substituting their corresponding values. Thus, 𝑚 𝐴𝐵 = 𝑚 𝐶𝐷 2𝑥 + 13 = 3𝑥 − 2 2𝑥 − 3𝑥 = −2 − 13 −𝑥 = −15 𝑥 = 𝟏𝟓
- 13. Example 2: ….. If 𝑚 𝐴𝐵 = 2𝑥 + 13 ° and 𝑚 𝐶𝐷 = 3𝑥 − 2 ° ,find 𝑚 𝐴𝐵 and 𝑚 𝐶𝐷. Solution 2: 𝑚 𝐴𝐵 = 2𝑥 + 13 = 2 15 + 13 = 30 + 13 = 𝟒𝟑 = 43°
- 14. ACTIVITY 1 1. a radius bisected bisected a 4 inches chord 𝐴𝐵 at midpoint Q, what would be measure of 𝐴𝑄 and 𝑄𝐵? 2. In circle 𝑂, the radius 𝑂𝐴 passes through the chord 𝐵𝐶 at its midpoint 𝐷. What relationship can be said between 𝑂𝐴 and 𝐵𝐶? 3. if the chord 𝑆𝐵 whose apothem is 2 inches and whose arc is 75⁰, is said to congruent with 𝐶𝐸, what would the measure of the apothem and arc of 𝐶𝐸?
- 15. ACTIVITY 2
- 16. ACTIVITY 3. Solve the following word problems.
- 17. ACTIVITY 3. Solve the following word problems.