Angle Measures and Segment Lengths

homework Inscribed angles page 5

Secants A secant is a line that intersects a circle at two points.

Angle Measures Theorem 12-13: 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 12-14: 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.

Find the measure of arc x. Ex.1 & 2: Find the mx. Find the measure of arc x. x° x° 92° 104° 68° 94° 268° 112° mx = ½(x - y) mx = ½(268 - 92) mx = ½(176) mx = 88° m1 = ½(x + y) 94 = ½(112 + x) 188 = (112 + x) 76° = x

Finding Angle Measures What is the value of each variable?

Finding Angle Measures Find the measure of the variable.

Angle Measures and Segment Lines Find the value of the variable. a. 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 11-11 (2)). 1 2 x = 88 Simplify. 11-4

Angle Measures and Segment Lines (continued) b. 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 11-11 (1)). 1 2 94 = x + 56 Distributive Property 1 2 38 = x Subtract. 1 2 76 = x Multiply each side by 2.

What is the value of each variable?

Segment Lengths Segment Lengths Theorem: For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and circle. In other words… Inside: Outside: “part” x “part” “[outside] part” x “whole”

Find the value of the variable Find the value of the variable. If the answer is not a whole number, round to the nearest tenth.

Find the value of the variable Find the value of the variable. If the answer is not a whole number, round to the nearest tenth.

Ex. 3 & 4 Find the length of g. Find length of x. t2 = y(y + z) 8 15 g 3 x 7 5 t2 = y(y + z) 152 = 8(8 + g) 225 = 64 + 8g 161 = 8g 20.125 = g a•b = c•d (3)•(7) = (x)•(5) 21 = 5x 4.2 = x

Ex.5: 2 Secants Find the length of x. w(w + x) = y(y + z) 20 w(w + x) = y(y + z) 14(14 + 20) = 16(16 + x) (34)(14) = 256 + 16x 476 = 256 + 16x 220 = 16x 13.75 = x 14 16 x

Ex.6: A little bit of everything! Find the measures of the missing variables Solve for k first. w(w + x) = y(y + z) 9(9 + 12) = 8(8 + k) 186 = 64 + 8k k = 15.6 12 k 175° 9 8 60° Next solve for r t2 = y(y + z) r2 = 8(8 + 15.6) r2 = 189 r = 13.7 a° r Lastly solve for ma m1 = ½(x - y) ma = ½(175 – 60) ma = 57.5°

Angle Measures and Segment Lines Find the value of the variable. a. 5 • x = 3 • 7 Along a line, the product of the lengths of two segments from a point to a circle is constant (Theorem 11-12 (1)). 5x = 21 Solve for x. x = 4.2 b. 8(y + 8) = 152 Along a line, the product of the lengths of two segments from a point to a circle is constant (Theorem 11-12 (3)). 8y + 64 = 225 Solve for y. 8y = 161 y = 20.125 11-4

Angle Measures and Segment Lines 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. The perpendicular bisector of the chord AB contains the center of the circle. 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. x • x = 50 • 200 Along a line, the product of the lengths of the two segments from a point to a circle is constant (Theorem 11-12 (1)). x2 = 10,000 Solve for x. x = 100 The shortest distance from point A to point B is 200 ft.

Angle Measures and Segment Lines Use M for Exercises 1 and 2. 1. Find a. 2. Find x. . Use O for Exercises 3–5. 3. Find a and b. 4. Find x to the nearest tenth. 5. Find the diameter of O. . 82 a = 60; b = 28 24 15.5 . 22

Finding Segment Lengths Find the value of the variable.

 What is the value of the variable to the nearest tenth?