Chapter 10 Circles 10.1 Properties of Tangents Circle: the set of all points in a plane that are equidistant from a given point. (Center) A circle is named by its center. (show symbol on board.) Radius: A segment from the center of a circle to any point on the circle. Chord: A segment whose endpoints are on the circle. Diameter: a chord containing the center of the circle. Diameter = two times radius. Diameter = 2 times Radius. Lines in circles: Secants: a line that intersects a circle at two points. Tangents: a line that intersects a circle in exactly one point. Point of tangency Tell whether the line, ray, or segment is best described as a B radius, chord, A diameter, secant, or C E tangent of circle C. D a. Segment BC b. Line DA c. Line DE a. Radius b. Tangent c. Secant Common Tangents Common internal tangents Common external tangents Tell how many common tangents the circles have and draw them. A. B. C. D. E. A. 3 B. 2 C. 1 D. 4 E. 0 Geometry CP Page 655(1-17, 27, 28, 35, 36) Sophomore Math Page 655(1-17, 35, 36) Two Tangent Rules (theorems) 1. In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle. Two Tangent Rules • Tangent segments from an external point are congruent. In the diagram, RS is a radius of circle R. Is ST tangent to circle R? 24 10 26 Use the converse of the pythagorean theorem to find out whether it is a right triangle. 26² = 10² + 24² That proves that it is a right triangle making the line perpendicular to the radius. Therefore, it must be a tangent. QR is tangent to the circle at R and QS is tangent at point S. Find the value of x. R 32 C Q 3x+5 S QR + QS Tangent segments from the same point are congruent. 32= 3x + 5 27 = 3x 9 = x EXAMPLE 2 Find lengths in circles in a coordinate plane Use the diagram to find the given lengths. a. Radius of A b. Diameter of A c. Radius of B d. Diameter of B SOLUTION a. The radius of A is 3 units. b. The diameter of A is 6 units. c. The radius of B is 2 units. d. The diameter of B is 4 units. EXAMPLE 4 Verify a tangent to a circle In the diagram, PT is a radius of P. Is ST tangent to P ? SOLUTION Use the Converse of the Pythagorean Theorem. Because 122 + 352 = 372, PST is a right triangle and ST PT . So, ST is perpendicular to a radius of P at its endpoint on P. By Theorem 10.1, ST is tangent to P. EXAMPLE 5 Find the radius of a circle In the diagram, B is a point of tangency. Find the radius r of C. SOLUTION You know from Theorem 10.1 that AB BC , so ABC is a right triangle. You can use the Pythagorean Theorem. AC2 = BC2 + AB2 Pythagorean Theorem (r + 50)2 = r2 + 802 Substitute. r2 + 100r + 2500 = r2 + 6400 Multiply. 100r = 3900 Subtract from each side. r = 39 ft . Divide each side by 100. EXAMPLE 6 Find the radius of a circle RS is tangent to C at S and RT is tangent to C at T. Find the value of x. SOLUTION RS = RT Tangent segments from the same point are 28 = 3x + 4 Substitute. 8 = x Solve for x. Geometry CP Page 656(18-26,29,33,37,43-47) Sophomore Math Page 656(18-20,24,27,28,30,37,43-47) .