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Properties of Chords and Arcs. Part 1. Definition of Chord. A chord is a segment that has its endpoints on a circle. An arc is part of a circle. P. Q. 6. r. 4. T. r. S. Review Example. PQ is tangent to circle S at P. Find the length of a radius r. Use the Pythagorean Theorem!.
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Definition of Chord • A chord is a segment that has its endpoints on a circle. • An arc is part of a circle.
P Q 6 r 4 T r S Review Example PQ is tangent to circle S at P. Find the length of a radius r. Use the Pythagorean Theorem! PS2 + PQ2 = SQ2 r2 + 62 = (r + 4)2 r2 + 36 = r2 + 8r + 16 36 = 8r + 16 20 = 8r r = 20/8 = 2.5
E B 30 in C 8 in 14 in D A Example 2 A belt fits around two pulleys. Find the distance between the pulleys. AE2 + ED2 = AD2 (14-8)2 + 302 = AD2 62 + 302 = AD2 AD2 = 936 AD = 30.6 in
P Q 15 in r 9 in T r O Example 3 PQ is tangent to circle O at P. Find the length of a radius r. PO2 + PQ2 = OQ2 r2 + 152 = (r + 9)2 r2 + 225 = r2 + 18r + 81 225 = 18r + 81 144 = 18r r = 144/18 = 8 in
Example 4 Is LM tangent to circle N at L? L M 24 If LM is tangent to circle N, then LMN is a right triangle. 25 N 72 + 242 = 49 + 576 = 625 625 = 252 Therefore, LMN is a right triangle
Circumscribing Circles The sides of the triangle are tangent to the circle. The triangle is circumscribed about the circle. The circle is inscribed in the triangle.
Theorem 12-4: If AB and CB are tangent to circle Ο at A and C, then AB CB Circumscribing Circles A O B C
A 10 cm D 15 cm F O B 8 cm E C Example 5 Circle O is inscribed in ABC. Find the perimeter of ABC. AD = AF = 10 cm BD = BE = 15 cm CF = CE = 8 cm The perimeter is AD + AF + BD + BE + CF + CE perimeter = 10 + 10 + 15 + 15 + 8 + 8 = 66 cm