1. GEOMETRY HELP x = (268 – 92)The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs (Theorem 12-11 (2)). 1212 x = 88Simplify. a. Find the value of the variable. Angle Measures and Segment Lengths LESSON 12-4 Additional Examples

2. GEOMETRY HELP 94 = (x + 112)The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs (Theorem 12-11 (1)). 1212 76 = xMultiply each side by 2. b. (continued) 94 = x + 56Distributive Property 1212 38 = xSubtract. 1212 Quick Check Angle Measures and Segment Lengths LESSON 12-4 Additional Examples

3. GEOMETRY HELP An advertising agency wants a frontal photo of a “flying saucer” ride at an amusement park. The photographer stands at the vertex of the angle formed by tangents to the “flying saucer.” What is the measure of the arc that will be in the photograph? In the diagram, the photographer stands at point T. TX and TY intercept minor arc XY and major arc XAY. Angle Measures and Segment Lengths LESSON 12-4 Additional Examples

4. GEOMETRY HELP 72 = 180 – x Distributive Property x + 72 = 180 Solve for x. x = 108 A 108° arc will be in the advertising agency’s photo. 72 = [(360 – x) – x] Substitute. 1212 72 = (360 – 2x) Simplify. 1212 Then mXAY = 360 – x. Let mXY = x. (continued) m T = (mXAY – mXY)The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs (Theorem 12-11 (2)). 1212 Angle Measures and Segment Lengths LESSON 12-4 Additional Examples Quick Check

5. GEOMETRY HELP 5 x = 3 7Along a line, the product of the lengths of two segments from a point to a circle is constant (Theorem 12-12 (1)). 5x = 21 Solve for x. x = 4.2 Find the value of the variable. a. 8(y + 8) = 15 2 Along a line, the product of the lengths of two segments from a point to a circle is constant (Theorem 12-12 (3)). 8y + 64 = 225 Solve for y. 8y = 161 y = 20.125 b. Angle Measures and Segment Lengths LESSON 12-4 Additional Examples Quick Check

6. GEOMETRY HELP Because the radius is 125 ft, the diameter is 2 125 = 250 ft. The length of the other segment along the diameter is 250 ft – 50 ft, or 200 ft. A tram travels from point A to point B along the arc of a circle with a radius of 125 ft. Find the shortest distance from point A to point B. x x = 50 200Along a line, the product of the lengths of the two segments from a point to a circle is constant (Theorem 12-12 (1)). x 2 = 10,000 Solve for x. x = 100 The shortest distance from point A to point B is 200 ft. The perpendicular bisector of the chord AB contains the center of the circle. Quick Check Angle Measures and Segment Lengths LESSON 12-4 Additional Examples