10.3 Apply Properties of Chords Hubarth Geometry

Theorem 10.3 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. A B C D

In the diagram, P Q, FG JK, and mJK = 80 o. Find mFG Because FG and JK are congruent chords in congruent circles, the corresponding minor arcs FG and JK are congruent. So, mFG = mJK = 80 o. Ex 1 Use Congruent Chords to Find an Arc Measures

Theorem J M P K L

Theorem B F E D G

Use the diagram of E to find the length of AC. Tell what theorem you use. Diameter BD is perpendicular to AC. So, by Theorem 10.5, BD bisects AC, and CF = AF. Therefore, AC = 2 AF = 2(7) = 14. Ex 2 Use a Diameter

Theorem 10.6 In the same circle, or in congruent circles. two chords are congruent if and only if they are equidistant from the center.. A B C D G F E

Ex 3 Use Theorem 10.6 Chords QR and ST are congruent, so by Theorem 10.6 they are equidistant from C. Therefore, CU = CV. CU = CV 2x = 5x – 9 x = 3 So, CU = 2x = 2(3) = 6. In the diagram of C, QR = ST = 16. Find CU.

Practice Use the diagram of D. 1. If mAB = 110 °, find mBC mBC = 110 ° 2. If mAC = 150 °, find mAB mAB = 105 ° Find the measure of the indicated arc. 3. CD 4. DE 5. CE mCD = 72° mDE = 72° mCE = 72° + 72° = 144° Find the given length 6. QR 7. QU 8. The radius of circle C r = 20