6 3 Apply Properties of Chords Theorem 6

6. 3 Apply 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. B C A D

6. 3 Apply Properties of Chords Example 1 Use congruent chords to find an arc measure E B In the diagram, A D, BC EF, and m. EF = 125 o. D Find m. BC. F A Solution C Because BC and EF are congruent ______ chords in circles the corresponding minor arcs congruent _______, congruent BC and EF are _____.

6. 3 Apply Properties of Chords Theorem 6. 6 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. T S P Q R If QS is a perpendicular bisector of TR, then ____ is a diameter of the circle.

6. 3 Apply 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 E H G D If EG is a diameter and EG HD HF and ____. DF, then

6. 3 Apply Properties of Chords Example 2 Use perpendicular bisectors 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 Label the sculptures A, B, and C. Draw segments AB and BC perpendicular bisectors of AB and BC. Step 2 Draw the __________ By _______, these bisectors are Theorem 6. 6 diameters of the circle containing A, B, and C. intersect Step 3 Find the point where these bisectors _____. This is the center of the circle containing A, B, and C, and so it is _____ equidistant from each point.

6. 3 Apply Properties of Chords Checkpoint. Complete the following exercises. S T 1. If m. TV = 121 o, find m. RS R V By Theorem 6. 5, the arcs are congruent. m. RS = 121 o

6. 3 Apply Properties of Chords Checkpoint. Complete the following exercises. C 2. Find the measures of CB, BE, and CE. B D E By Theorem 6. 7, the diameter bisects the chord. m. CB = 64 o m. BE = 64 o m. CE = 128 o

6. 3 Apply 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. C A G E F D B AB CD if and only if ____.

6. 3 Apply Properties of Chords Example 3 Use Theorem 6. 8 In the diagram of F, AB = CD = 12. Find EF. Solution A G B F D E C Chords AB and CD are congruent, so equidistant by Theorem 6. 8 they are _____ GF from F. Therefore, EF = _____. Use Theorem 6. 8. Substitute. Solve for x. 6 So, EF = 3 x = 3(___) 2 = ___.

6. 3 Apply Properties of Chords Checkpoint. Complete the following exercises. A 3. In the diagram in Example 3, suppose AB = 27 and EF = GF = 7. Find CD. B G F D E C By Theorem 6. 8, the two chords are congruent since they are equidistant from the center. CD = 27

6. 3 Apply Properties of Chords Example 4 Use chords with triangle similarity In S, SP = 5, MP = 8, ST = SU, QN M MP, and NRQ is a right angle. U Show that PTS NRQ. R 1. Determine the side lengths of PTS. Diameter QN is perpendicular to MP, Theorem 6. 7 QN bisects MP. Therefore, so by ______ N T P S Q SP has a given length of ___. Because QN is perpendicular to MP, The side lengths of PTS are SP = ____, PT = ____, and TS = ____. right angle PTS is a _____

6. 3 Apply Properties of Chords Example 4 Use chords with triangle similarity In S, SP = 5, MP = 8, ST = SU, QN M MP, and NRQ is a right angle. U Show that PTS NRQ. 2. Determine the side lengths of NRQ. R 5 so The radius SP has a length of ___, SP = 2(__) 10 the diameter QN = 2(___) 5 = ___. 8 Theorem 6. 8 NR MP, so NR = MP = __. By _______ right angle Because NRQ is a ______, The side lengths of NRQ are QN = ___, NR = ___, and RQ = ___. N T S Q P

6. 3 Apply 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. N T M U S R 3. Find the ratios of corresponding sides. Because the side lengths are proportional, PTS Side-Side Similarity Theorem by the ________________. Q NRQ P

6. 3 Apply 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. 1. NR = MP = 1. 24 then TP = 12 1. Since QN is the diameter and SP is 1. a radius, then SP = 13 N T M U S R Because the side lengths are proportional, PTS Side-Side Similarity Theorem by the ________________. Q NRQ P

6. 3 Apply Properties of Chords Pg. 211, 6. 3 #1 -26