10.5 Segment Length in Circles Geometry 10.5 Segment Length in Circles mbhaub@mpsaz.org

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

Quadratic Equation Review Skip A quadratic equation is in the form Any quadratic equation can be solved using the formula Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Solve: Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles

Geometry 10.5 Segment Lengths in Circles Solve: 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 Theorem Demo d a 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 Theorem Demo 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) Theorem Demo 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

Geometry 10.5 Segment Lengths in Circles Homework mbhaub@mpsaz.org Tuesday, March 24, 2:56 Geometry 10.5 Segment Lengths in Circles