Other Angle Relationships in Circles Geometry Other Angle Relationships in Circles

Goal Use angles formed by tangents, secants, and chords to solve problems. 7/1/2018

Review Note: in solving an equation with fractions, one of the first things to do is always “clear the fractions”. 7/1/2018

You do it. Solve: 7/1/2018

Review The measure of an inscribed angle is equal to one-half the measure of the intercepted arc. 40 80 What if one side of the angle is tangent to the circle? 7/1/2018

Inside the circle. On the circle. Outside the circle. If two lines intersect a circle, where can the lines intersect each other? Inside the circle. We already know how to do this. On the circle. Outside the circle. 7/1/2018

Theorem 12.13 (Inside the circle) If two chords intersect in a circle, then the measure of the angle is one-half the sum of the intercepted arcs. A C 1 B D 7/1/2018

Simplified Formula 1 a b 7/1/2018

Example 3 Find m1. A C 30 1 B 80 D 7/1/2018

Example 4 Solve for x. 20 x A C 60 B 100 Check: 100 + 20 = 120 120 ÷ 2 = 60 D 7/1/2018

Your turn. Solve for x & y. 20 75 85 M y K P O 32 A B C D x 7/1/2018

Intersection Outside the Circle 7/1/2018

Secant-Secant A C 1 D B 7/1/2018

Simplified Formula 1 b a 7/1/2018

Secant-Tangent A C 1 B 7/1/2018

Simplified Formula a b 1 7/1/2018

Tangent-Tangent A C 1 B 7/1/2018

Simplified Formula 1 a b 7/1/2018

Intersection Outside the Circle Secant-Secant Secant-Tangent Tangent-Tangent In all cases, the measure of the exterior angle is found the same way: One-half the difference of the larger and smaller arcs. (Click the titles above for Sketchpad Demonstrations.) 7/1/2018

Example 5 Find m1. 80 35 1 10 7/1/2018

Example 6 Find m1. 120 70 1 25 7/1/2018

Example 7 Find m1. k 210 ? 150 30 1 m 360 – 210 = 150 Rays k and m are tangent to the circle. 210 ? 150 30 1 m 360 – 210 = 150 7/1/2018

How to remember this: If the angle vertex is on the circle, its measure is one-half the intercepted arc. If an angle vertex is inside the circle, its measure if one-half the sum of the intercepted arcs. If an angle vertex is outside the circle, its measure is one-half the difference of the intercepted arcs. 7/1/2018

Chords in a Circle Theorem 12.15 b c a  b = c  d 7/1/2018

Example 1 Find a. 10  4 = 8  a 40 = 8  a 5 = a 10 5 a 4 8 7/1/2018

Your Turn: Find x. 3x  x = 8  6 3x2 = 48 x2 = 16 3x 6 x = 4 12 x 4 8 A D 3x 6 12 E x 4 B 8 C Check: 12  4 = 48 and 8  6 = 48 7/1/2018

Terminology This line is a secant. This segment is a secant segment. 7/1/2018

Terminology This segment is the external secant segment. 7/1/2018

Terminology This line is a tangent. This segment is a tangent segment. 7/1/2018

Terminology AC is a __________________. secant segment AB is the _________________________. AD is a _________________. secant segment external secant segment tangent segment A B C D 7/1/2018

Theorem 12.17 (tangent-secant) B C D 7/1/2018

Theorem 12.17 (simplified) c2 = a(a + b) b a c 7/1/2018

Example 2 Find AD. A B C D 6 4 7/1/2018

Your Turn. Solve for x. 8 4 4 x 7/1/2018

Turn it up a notch… 4 x Now What? 5 7/1/2018

Quadratic Equation Set quadratic equations equal to zero. 7/1/2018

Quadratic Formula 1 a = 1 b = 4 c = -25 7/1/2018

Quadratic Formula 1 a = 1 b = 4 c = -25 7/1/2018

Solve it. x can’t be negative x  3.39 7/1/2018

All that for just one problem? Just do it! 7/1/2018

Your Turn Solve for x. 3 2 x Equation: 32 = x(x + 2) x + 2 7/1/2018

Solution 32 = x(x + 2) 9 = x2 + 2x 0 = x2 + 2x – 9 a = 1 b = 2 c = -9 7/1/2018

Theorem 12.16 (secant-secant) b a d c a(a+b) = c(c+d) 7/1/2018

Example 3 Solve for x. Solution: 5(5 + 8) = 6(6 + x) 5(13) = 36 + 6x x = 4 5/6 (or 4.83) 5 8 6 X 7/1/2018

Your Turn Solve for x. 9 11 10 X 7/1/2018

Solution 9 11 10 X 7/1/2018

Example 4 Solve for x. Equation: 5x = 4(16) Why? 5x = 64 x = 12.8 4 16 7/1/2018

Formula Summary a(a+b) = c(c+d) a b c d c a b c2 = a(a + b) a c d b 7/1/2018

Practice Problems 7/1/2018