Tell whether the segment is best described as a radius, chord, or diameter of C. 1. DC radius 2. BD diameter 3. DE chord 4. AE chord Solve 5. 4x = 8x – 12. x = 3 6. 3x + 2 = 6x – 4. x = 2

Use properties of segments that intersect circles. Target Use properties of segments that intersect circles. You will… Use relationships of arcs and chords in a circle.

Vocabulary Chords and Arcs Theorem 10.3 - In the same circle or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Theorem 10.6 - In the same circle or congruent circles, two chords are congruent if and only if they are equidistant from the center.

Vocabulary Chords and Arcs Theorem 10.4 – If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. Theorem 10.5 – If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

EXAMPLE 1 Use congruent chords to find an arc measure In the diagram, P Q, FG JK , and mJK = 80o. Find mFG . SOLUTION Because FG and JK are congruent chords in congruent circles, the corresponding minor arcs FG and JK are congruent. So, mFG = mJK = 80o.

GUIDED PRACTICE for Example 1 Use the diagram of D. 1. If mAB = 110°, find mBC mBC = 110° ANSWER 2. If mAC = 150°, find mAB mAB = 105° ANSWER

EXAMPLE 2 Use perpendicular bisectors Three bushes are arranged in a garden as shown. Where should you place a sprinkler so that it is the same distance from each bush? Gardening SOLUTION STEP 1 Label the bushes A, B, and C, as shown. Draw segments AB and BC .

EXAMPLE 2 Use perpendicular bisectors STEP 2 Draw the perpendicular bisectors of AB and BC By Theorem 10.4, these are diameters of the circle containing A, B, and C. STEP 3 Find the point where these bisectors intersect. This is the center of the circle through A, B, and C, and so it is equidistant from each point.

EXAMPLE 3 Use a diameter 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. ANSWER

GUIDED PRACTICE for Examples 2 and 3 Find the measure of the indicated arc in the diagram. mCD = 72° ANSWERS 3. CD 4. DE mCD = mDE. mDE = 72° 5. CE mCE = mDE + mCD mCE = 72° + 72° = 144°

In the diagram of C, QR = ST = 16. Find CU. EXAMPLE 4 Use Theorem 10.6 In the diagram of C, QR = ST = 16. Find CU. SOLUTION Chords QR and ST are congruent, so by Theorem 10.6 they are equidistant from C. Therefore, CU = CV. CU = CV Use Theorem 10.6. 2x = 5x – 9 Substitute. x = 3 Solve for x. So, CU = 2x = 2(3) = 6.

GUIDED PRACTICE for Example 4 In the diagram in Example 4, suppose ST = 32, and CU = CV = 12. Find the given length. QR = 32 ANSWERS 6. QR 7. QU QU = 16 8. The radius of C r = 20