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MM2G3a

Standard. MM2G3a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Theorem 6.5. In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

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MM2G3a

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  1. Standard MM2G3a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

  2. Theorem 6.5 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

  3. In the diagram, PQ, FGJK, and mJK= 80o. Find mFG Because FGand JKare congruent chords in congruent circles, the corresponding minor arcs FGand JKare congruent. So, mFG = mJK = 80o. EXAMPLE 1 Use congruent chords to find an arc measure SOLUTION

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

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

  6. Theorem 6.6 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. If BD is a perpendicular bisector of EC, then BD is a diameter of the circle.

  7. Gardening 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? Label the bushes A, B, and C, as shown. Draw segments ABand BC. STEP 1 EXAMPLE 2 Use perpendicular bisectors SOLUTION

  8. Draw the perpendicular bisectors of ABand BC.By Theorem 6.6, these are diameters of the circle containing A, B, and C. STEP 2 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 2 Use perpendicular bisectors

  9. Theorem 6.7 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

  10. Use the diagram of Eto find the length of AC. Tell what theorem you use. ANSWER Diameter BDis perpendicular to AC. So, by Theorem 6.7, BDbisects AC, and CF = AF. Therefore, AC = 2 AF =2(7) = 14. EXAMPLE 3 Use a diameter

  11. Find the measure of the indicated arc in the diagram. 3. CD ANSWER mCD=72° for Examples 2 and 3 GUIDED PRACTICE

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

  13. Theorem 6.8 In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

  14. In the diagram of C, QR = ST = 16. Find CU. Chords QRand STare congruent, so by Theorem 6.8 they are equidistant from C. Therefore, CU = CV. EXAMPLE 4 SOLUTION CU = CV Use Theorem 6.8 2x = 5x – 9 Substitute. Solve for x. x = 3 So, CU = 2x = 2(3) = 6.

  15. In the diagram in Example 4, suppose ST = 32, and CU= CV = 12. Find the given length. ANSWER QR= 32 for Example 4 GUIDED PRACTICE 6. QR

  16. In the diagram in Example 4, suppose ST = 32, and CU= CV = 12. Find the given length. ANSWER QU= 16 for Example 4 GUIDED PRACTICE 7. QU

  17. In the diagram in Example 4, suppose ST = 32, and CU= CV = 12. Find the given length. 8. The radius of C ANSWER The radius of C = 20 for Example 4 GUIDED PRACTICE

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