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6.3

6.3. Apply Properties of Chords. C. B. D. A. 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. 6.3. Apply Properties of Chords.

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6.3

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  1. 6.3 Apply Properties of Chords C B D A 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.

  2. 6.3 Apply Properties of Chords In the diagram, A D, BC EF, and mEF = 125o. Find mBC. Because BC and EF are congruent ______ in congruent _______, the corresponding minor arcs BC and EF are __________. E B D F A C Use congruent chords to find an arc measure Example 1 Solution chords circles congruent

  3. 6.3 Apply Properties of Chords T If QS is a perpendicular bisector of TR, then ____ is a diameter of the circle. S Q P R Theorem 6.6 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

  4. 6.3 Apply Properties of Chords F If EG is a diameter and EG DF, then HD HF and ____ ____. E G H D Theorem 6.7 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

  5. 6.3 Apply Properties of Chords Label the sculptures A, B, and C. Draw segments AB and BC Draw the ____________________ of AB and BC. By _____________, these bisectors are diameters of the circle containing A, B, and C. Use perpendicular bisectors Example 2 Journalism A journalist is writing a story about three sculptures, arranged as shown at the right. Where should the journalist place a camera so that it is the same distance from each sculpture? A C Solution B Step 1 perpendicular bisectors Step 2 Theorem 6.6 intersect Find the point where these bisectors _________. This is the center of the circle containing A, B, and C, and so it is __________ from each point. Step 3 equidistant

  6. 6.3 Apply Properties of Chords S T R V Checkpoint. Complete the following exercises. • If mTV = 121o, find mRS By Theorem 6.5, the arcs are congruent. mRS = 121o

  7. 6.3 Apply Properties of Chords C B D E Checkpoint. Complete the following exercises. • Find the measures ofCB, BE, and CE. By Theorem 6.7, the diameter bisects the chord. mCB = 64o mBE = 64o mCE = 128o

  8. 6.3 Apply Properties of Chords C A G AB CD if and only if ____ ____. E F D B 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.

  9. 6.3 Apply Properties of Chords In the diagram of F, AB = CD = 12. Find EF. Chords AB and CD are congruent, so by Theorem 6.8 they are __________ from F. Therefore, EF = _____. B A G F D E C Use Theorem 6.8 Example 3 Solution equidistant GF Use Theorem 6.8. Substitute. Solve for x. 6 2 So, EF = 3x = 3(___) = ___.

  10. 6.3 Apply Properties of Chords B A G F D E C Checkpoint. Complete the following exercises. • In the diagram in Example 3, suppose AB = 27 and EF = GF = 7. Find CD. By Theorem 6.8, the two chords are congruent since they are equidistant from the center. CD = 27

  11. 6.3 Apply Properties of Chords In S, SP = 5, MP = 8, ST = SU, QN MP, and NRQ is a right angle. Show that PTS NRQ. • Determine the side lengths of PTS. Diameter QN is perpendicular to MP, so by ___________ QN bisects MP. Therefore, N T M P U S Because QN is perpendicular to MP, PTS is a __________ R Q The side lengths of PTS are SP = ____, PT = ____, and TS = ____. Use chords with triangle similarity Example 4 Theorem 6.7 SP has a given length of ___. right angle

  12. 6.3 Apply Properties of Chords In S, SP = 5, MP = 8, ST = SU, QN MP, and NRQ is a right angle. Show that PTS NRQ. • Determine the side lengths of NRQ. The radius SP has a length of ___, so the diameter QN = 2(___) = 2(__) = ___. N T M P U By _____________ NR MP, so NR = MP = __. Because NRQ is a ____________, S R Q The side lengths of NRQ are QN = ___, NR = ___, and RQ = ___. Use chords with triangle similarity Example 4 5 SP 10 5 8 Theorem 6.8 right angle

  13. 6.3 Apply Properties of Chords In S, SP = 5, MP = 8, ST = SU, QN MP, and NRQ is a right angle. Show that PTS NRQ. N T M P U S Because the side lengths are proportional, PTS NRQ R by the ________________________________. Q Use chords with triangle similarity Example 4 • Find the ratios of corresponding sides. Side-Side-Side Similarity Theorem

  14. 6.3 Apply Properties of Chords • In Example 4, suppose in S, QN = 26, NR = 24, ST = SU, QN MP, and NRQ is a right angle. Show that PTS NRQ. N T M P U S Because the side lengths are proportional, PTS NRQ R by the ________________________________. Q Checkpoint. Complete the following exercises. then TP = 12 NR = MP = 24 Since QN is the diameter and SP is a radius, then SP = 13 Side-Side-Side Similarity Theorem

  15. 6.3 Apply Properties of Chords Pg. 211, 6.3 #1-26

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