10.5 Segment Length in Circles Geometry 10.5 Segment Length in Circles

Geometry 10.5 Segment Lengths in Circles Goals Find the lengths of segments of chords. Find the lengths of segments and tangents. Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles 10.5 Chords in a Circle Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Chords in a Circle Theorem 10.15 b c a  b = c  d Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Example 1 Find a. 10  4 = 8  a 40 = 8  a 5 = a 10 5 a 4 8 Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Your Turn: Find x. 3x  x = 8  6 3x2 = 48 x2 = 16 x = 4 A D 3x 6 12 E x 4 B 8 C Check: 12  4 = 48 and 8  6 = 48 Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Proof 3 and 4 both intercept arc SV. What does this tell use about 3 and 4? They are congruent. What kind of angles are 1 and 2? Vertical Angles And vertical angles are ____. Congruent. R T 3 O 4 1 2 V S Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Proof continued Now SOR ~ VOT. Why? AA~ Postulate. In similar triangles, sides are proportional: R T 3 O 4 1 2 V OR  OV = OT  OS S Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Terminology This line is a secant. This segment is a secant segment. Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Terminology This segment is the external secant segment. Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Terminology This line is a tangent. This segment is a tangent segment. Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Terminology AC is a __________________. AB is the _________________________. AD is a _________________. secant segment external secant segment tangent segment A B C D Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Theorem 10.17 (tangent-secant) B C D Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Theorem 10.17 (simplified) c2 = a(a + b) b a c Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Example 2 Find AD. A B C D 6 4 Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Your Turn. Solve for x. 8 4 4 x Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Turn it up a notch… 4 x Now What? 5 Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Quadratic Equation Set quadratic equations equal to zero. Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Quadratic Formula 1 a = 1 b = 4 c = -25 Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Quadratic Formula 1 a = 1 b = 4 c = -25 Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Solve it. x can’t be negative x  3.39 Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles All that for just one problem? Just do it! Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Your Turn Solve for x. 3 2 x Equation: 32 = x(x + 2) x + 2 Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles 3 2 x Solution 32 = x(x + 2) 9 = x2 + 2x 0 = x2 + 2x – 9 a = 1 b = 2 c = -9 Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Theorem 10.16 (secant-secant) b a d c a(a+b) = c(c+d) Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Example 3 Solve for x. Solution: 5(5 + 8) = 6(6 + x) 5(13) = 36 + 6x 65 = 36 + 6x 29 = 6x x = 4 5/6 (or 4.83) 5 8 6 X Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Your Turn Solve for x. 9 11 10 X Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Solution 9 11 10 X Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Example 4 Solve for x. Equation: 5x = 4(16) Why? 5x = 64 x = 12.8 16 12 5 X 4 Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Summary Segments in a circle have three situations: Chord-Chord Secant-Tangent Secant-Secant Do you know the formula for each? Read the problems carefully. Use the correct numbers for each variable. Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Formula Summary a(a+b) = c(c+d) a b c d c a b c2 = a(a + b) a c d b ab = cd Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles