6.3Apply Properties of Chords 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. C B A D

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

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

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

6.3Apply Properties of Chords Example 2 Use perpendicular bisectors Solution Label the sculptures A, B, and C. Draw segments AB and BC perpendicular bisectors Theorem 6.6 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 B C Step 1 Draw the ____________________ of AB and BC. By _____________, these bisectors are diameters of the circle containing A, B, and C. Step 2 Step 3 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. intersect equidistant

6.3Apply Properties of Chords Checkpoint. Complete the following exercises. 1.If mTV = 121 o, find mRS T S R V mRS = 121 o By Theorem 6.5, the arcs are congruent.

6.3Apply Properties of Chords Checkpoint. Complete the following exercises. 2.Find the measures of CB, BE, and CE. By Theorem 6.7, the diameter bisects the chord. C B E D mCB = 64 o mBE = 64 o mCE = 128 o

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

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

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

6.3Apply Properties of Chords Example 4 Use chords with triangle similarity Theorem 6.7 In S, SP = 5, MP = 8, ST = SU, QN MP, and NRQ is a right angle. Show that PTS NRQ. M T U N S R Q P 1.Determine the side lengths of PTS. Diameter QN is perpendicular to MP, so by ___________ QN bisects MP. Therefore, SP has a given length of ___. Because QN is perpendicular to MP, PTS is a __________ right angle The side lengths of PTS are SP = ____, PT = ____, and TS = ____.

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

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

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

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