Geometry Goals Know properties of circles. Identify special lines in a circle. Solve problems with special lines. March 23, 2016 Circle: Set of points on a plane equidistant from a point (center). B This is circle C, or C C AB is a diameter. R A March 23, 2016 CR is a radius. The diameter is twice the radius. Terminology One radius Two radii radii = ray-dee-eye March 23, 2016 All Radii in a circle are congruent March 23, 2016 Interior/Exterior A A is in the interior of the circle. C B March 23, 2016 C is on the circle. B is in the exterior of the circle. Congruent Circles Radii are congruent. March 23, 2016 March 23, 2016 Chord A chord is a segment between two points on a circle. A diameter is a chord that passes through the center. March 23, 2016 Secant A secant is a line that intersects a circle at two points. March 23, 2016 Tangent •A tangent is a line that intersects a circle at only one point. •It is called the point of tangency. March 23, 2016 Tangent Circles Intersect at exactly one point. These circles are externally tangent. March 23, 2016 Tangent Circles Intersect at exactly one point. These circles are internally tangent. March 23, 2016 Can circles intersect at two points? YES! March 23, 2016 Concentric Circles Have the same center, different radius. March 23, 2016 Concentric Circles Have the same center, different radius. March 23, 2016 Concentric Circles Have the same center, different radius. March 23, 2016 Concentric Circles Have the same center, different radius. March 23, 2016 Concentric Circles Have the same center, different radius. March 23, 2016 Concentric Circles Have the same center, different radius. March 23, 2016 Concentric Circles Have the same center, different radius. March 23, 2016 Concentric Circles Have the same center, different radius. March 23, 2016 Common External Tangents And this is a common external tangent. This is a common external tangent. March 23, 2016 Common External Tangents in a real application… March 23, 2016 Common Internal Tangents And this is a common internal tangent. This is a common internal tangent. March 23, 2016 March 23, 2016 Theorem 12.1 (w/o proof) If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. March 23, 2016 Theorem 12.2 (w/o proof) If a line drawn to a circle is perpendicular to a radius, then the line is a tangent to the circle. (The converse of 10.1) March 23, 2016 Example 1 Is RA tangent to T? R 12 5 13 T A YES 52 + 122 = 132 25 + 144 = 169 TA = 13 March 23, 2016 169 = 169 RAT is a right triangle. FOIL Find (x + 3)2 (x + 3)(x + 3) March 23, 2016 FOIL Find (x + 3)2 (x + 3)(x + 3) x2 March 23, 2016 FOIL Find (x + 3)2 3x (x + 3)(x + 3) x2 March 23, 2016 FOIL Find (x + 3)2 (x + 3)(x + 3) 3x x2 + 3x March 23, 2016 FOIL Find (x + 3)2 (x + 3)(x + 3) 9 x2 + 3x + 3x March 23, 2016 FOIL Find (x + 3)2 (x + 3)(x + 3) x2 + 3x + 3x + 9 March 23, 2016 FOIL (x + 3)2 = x2 + 6x + 9 March 23, 2016 Expand (x + 9)2 (x + 9)(x + 9) F: x2 O: 9x I: 9x L: 81 (x + 9)2 = x2 + 18x + 81 March 23, 2016 BC is tangent to circle A at B. Find r. Example 2 A r AC = r? + 16 D 16 r B 24 C DC = 16 r2 + 242 = (r + 16)2 March 23, 2016 Solve the equation. r2 + 242 = (r + 16)2 r2 + 576 = (r + 16)(r + 16) r2 + 576 = r2 + 16r + 16r + 256 576 = 32r + 256 320 = 32r r2 + 242 = (r + 16)2 r = 10 March 23, 2016 Here’s where the situation is now. A 10 26 D 16 10 B AC = 26 r = 10 March 23, 2016 24 Check: C 102 + 242 = 262 100 + 576 = 676 676 = 676 Theorem 12.3 If two segments from the same exterior point are tangent to a circle, then the segments are congruent. Theorem Demo March 23, 2016 Example 3 HE and HA are tangent to the circle. Solve for x. A 12x + 15 H 9x + 45 E March 23, 2016 Solution 12x + 15 = 9x + 45 3x + 15 = 45 12(10) + 15 A 120 + 15 = 135 12x + 15 3x = 30 H x = 10 E 9x + 45 9(10) + 45 90 + 45 = 135 March 23, 2016 Try This: The circle is tangent to each side of ABC. Find the perimeter of ABC. 7 + 12 + 9 = 28 A 2 2 9 7 7 C March 23, 2016 5 7 5 12 B Can you… Identify a radius, diameter? Recognize a tangent or secant? Define Concentric circles? Internally tangent circles? Externally tangent? Tell the difference between internal and external tangents? Solve problems using tangent properties? March 23, 2016 Practice Problem 1 MD and ME are tangent to the circle. Solve for x. 4x – 12 = 2x + 12 D 4x 12 2x – 12 = 12 M 2x = 24 x = 12 March 23, 2016 2x + 12 E Practice Problem 2 R x 4 T Solve for x. March 23, 2016 12 x2 + 42 = (4 + 12)2 x2 + 16 = 256 x2 = 240 x = 415 15.5 Practice Problem 3 R 8 x T x 6 x2 + 82 = (x + 6)2 x2 + 64 = x2 + 12x + 36 64 = 12x + 36 Solve for x. 28 = 12x x = 2.333… March 23, 2016 Practice Problems March 23, 2016