10.1—Properties of Tangents Circle— Radius— Tangent Chord Diameter Chord— Radius Diameter— Secant Secant— Tangent— Example— Tell whether the line or segment is best descrbed as a radius, chord, diameter, secant, or tangent of ๏C. F a. E b. A C B c. d. D Coplanar circles can intersect in two points, one point, or no points. Common tangent— How many common tangents do the circles have? Draw them. Theorem 10.1— In ๏G, GH is a radius. Is I HI tangent to ๏G? 75 72 G 21 H In ๏K, J is a point of tangency. Find the radius of ๏K. K r 36 cm r J 48 cm Theorem 10.2— Find x. 25 6x - 8 L 10.1 Homework 3-10 19. 24. 26. 13. 23. 25. 29. 34. 37. 10.2—Find Arc Measures Central angle— central angle Minor arc— Major arc— A minor arc AB B C Semicircle— D major arc ADB E Measuring Arcs— 65° F G Measure of a major arc— Find the measure of each arc of ๏K where HJ is a diameter. I a. J b. 80° c. d. H K Example—a result of a survey about the ages of people in a town are shown. Find the indicated arc measures. Ages of People (in years) T S a. mRU b. mRST c. mRVT d. mUST >65 17-44 100° 90° U Q 80° 60° 45-64 R 15-17 V Congruent circles— Congruent arcs— Tell whether arcs CD and EF are congruent. Why? a. D b. E c. D E C 45° 45° P C F 110° Q C P D F E F 10.2 HW PICTURES: 3 – 10 17. 21. 20. 10.3—Properties of Chords Theorem 10.3— Theorem 10.4— Theorem 10.5— A Examples— C 1. In ๏R mAB = _______˚. Find the mCD. 108° B 2. Use the diagram of ๏C to find the length of BF . 3. In the diagram of ๏P, PV = PW, QR = 2x + 6, and ST = 3x – 1. Find QR. R A D 15 F C B V S Q P W G D T 17. 21. 23. 30. 10.4—Inscribed angles and Polygons Inscribed angle— A C D B Intercepted arc— Theorem 10.7—Measure of an Inscribed Angle Theorem— A C Theorem 10.8— D E B Theorem 10.9— R S T --Conversely, Q Theorem 10.10— Examples— 1. Find the indicated measures in ๏X a. mUW b. m V 104° ∠ VWY 33° X U 2. Find mWX and m Y W ∠ WYX. X W 44° Z Find the measure of each angle in the quadrilateral. 3. + 5)° (4y 5x 7x (5y - 5)° A Y 10.5—Other Angle Relationships in Circles A Theorem 10.11— D 1 2 Find the indicated measures. B C A mAB = 124° D 1 2 B C Intersecting lines and circles—if two lines intersect a circle, there are three places where the lines can intersect Theorem 10.12—Angles Inside the Circle— D E 1 2 F G Theorem 10.13—Angles Outside the Circle— Examples— Line m is tangent to the circle. Find x or y. 1. 2. 228° y° x° 118° 3. 4. x° 89° x 63° 44° 30° 5. 39° (7x - 2)° (17x + 6)° 10.6—Finding Segment Lengths in Circles Theorem 10.14—Segments of Chords— C A E D B Find x. Secant segment— External segment— Theorem 10.15—Segments of Secants Theorem— Find x. x Find RT. T 21 x R V 27 10.16—Segments of Secants and Tangents Theorem— Find x. x 24 x x-4 10.7—Write and Graph Equations of Circles 6 Standard Equation of a Circle Centered at the Origin: 4 2 -10 -5 5 10 -2 -4 -6 -8 Standard Equation of a Circle Centered at (h, k): 3 2 (h, k) 1 -4 -2 2 -1 -2 Write the equations of the circles using the given information. 1. Center (0, 0), radius 5 2. Center (-3, 8), radius 5/3 3. Center (1, 2) and a point on the circle is (4, 2) 4 Graph each equation. 4. x2 + y2 = 25 5. (x - 2)2 + (y + 1)2 = 4