1. Arcs and Chords

2. A chord is a segment with endpoints on a circle. If a chord is not a diameter, then its endpoints divide the circle into a major and minor arc.

3. Example 1: a) A circular piece of jade is hung from a chain by two wires around the stone.

4. Example 1:

5. Example 2: WX= YZDefinition of congruent segments 7x – 2= 5x + 6Substitution 7x= 5x + 8Add 2 to each side. 2x= 8Subtract 5x from each side. x= 4Divide each side by 2. So, WX = 7x – 2 = 7(4) – 2 or 26.

6. Example 2: RT= LMDefinition of congruent segments 3x – 5= 2x + 1Substitution 3x= 2x + 6Add 5 to each side. x= 6Subtract 2x from each side. So, LM = 2x + 1 = 2(6) + 1 or 13.

8. Example 3:

10. Example 4: a) In the ceramic stepping stone below, diameter AB is 18 inches long and chord EF is 8 inches long. Find CD. Draw radius CE. This forms right ΔCDE.

11. Find CE and DE. Since AB = 18 inches, CB = 9 inches. All radii of a circle are congruent, so CE = 9 inches. Since diameter AB is perpendicular to EF, AB bisects chord EF by Theorem 10.3. So, DE = 0.5(8) or 4 inches. Use the Pythagorean Theorem to find CD. CD 2 + DE 2 = CE 2 Pythagorean Theorem CD 2 + 4 2 = 9 2 Substitution CD 2 + 16 = 81Simplify. CD 2 = 65Subtract 16 from each side. Take the positive square root.

12. Example 4: b) In the circle below, diameter QS is 14 inches long and chord RT is 10 inches long. Find VU. Draw radius VR. This forms right ΔVRU.

13. Find VR and UR. Since QS = 14 inches, VS = 7 inches. All radii of a circle are congruent, so VR = 7 inches. Since diameter QS is perpendicular to RT, QS bisects chord RT by Theorem 10.3. So, UR = 0.5(10) or 5 inches. Use the Pythagorean Theorem to find VU. VU 2 + UR 2 = VR 2 Pythagorean Theorem VU 2 + 5 2 = 7 2 Substitution VU 2 + 25 = 49Simplify. VU 2 = 24Subtract 25 from each side. Take the positive square root.

15. Example 5: Since chords EF and GH are congruent, they are equidistant from P. So, PQ = PR. PQ= PR 4x – 3= 2x + 3Substitution x= 3Simplify. So, PQ = 4(3) – 3 or 9

16. Example 5: RS = 15