Chapter 10 Properties of Circles

1. Chapter 10 Properties of Circles Mrs. Pullo SEPTEMBER 26, 2017

2. Circle • A set of all points in a plane that are equidistant from a given point (the center).

3. Radius • Segment whose endpoints are the center and any point on the circle.

4. Chord • A segment whose endpoints are on a circle.

5. Diameter • A chord that contains the center of a circle.

6. Secant • A line that intersects a circle in 2 points.

7. Tangent • A Line in a plane of a circle that intersects the circle in exactly one point, the point of tangency.

9. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. AC AC a. a. is a radius because Cis the center and Ais a point on the circle. EXAMPLE 1

10. is a diameter because it is a chord that contains the center C. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. b. AB AB b.

11. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. is a tangent ray because it is contained in a line that intersects the circle at only one point. c. c. DE DE

12. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. d. AE d. is a secant because it is a line that intersects the circle in two points. AE

13. Use the diagram to find the given lengths. a. a. The radius of Ais 3 units. Radius ofA b. b. The diameter of Ais 6 units. Diameter of A c. c. The radius of B is 2 units. Radius ofB Diameter ofB The diameter of Bis 4 units. d. d.

14. Theorem • 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.

15. In the diagram, PTis a radius of P. Is STtangent to P ? Use the Converse of the Pythagorean Theorem. Because 122 + 352 = 372,PSTis a right triangle and STPT. So, STis perpendicular to a radius of Pat its endpoint on P. By Theorem 10.1, STis tangent to P.

16. In the diagram, Bis a point of tangency. Find the radiusr of C. You know from Theorem 10.1 that AB BC, so ABCis 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.

17. Theorem • Tangent segments from a common external point are congruent.

18. RSis tangent to Cat Sand RTis tangent to Cat T. Find the value of x. Tangent segments from the same point are RS= RT 28 = 3x + 4 Substitute. 8 = x Solve for x.

19. 3. Find the value(s)of x. 1. IsDEtangent to C? +3= x r = 7 Yes 2. ST is tangent toQ.Find the value of r.

20. Vocabulary • Central Angle • An angle whose vertex is the center of the circle. • Minor Arc • If angle ACB is less than 180° • Major Arc • Points that do not lie on the minor arc. • Semi Circle • Endpoints are the diameter

21. Measures • Measure of a Minor Arc • The measure of it’s central angle. • Measure of a Major Arc • Difference between 360 and the measure of the minor arc.

22. Find the measure of each arc of P, where RTis a diameter. c. a. RTS RST RS b. a. RSis a minor arc, so mRS=mRPS=110o. b. – RTSis a major arc, so mRTS = 360o 110o = 250o. c. RT is a diameter, so RSTis a semicircle, and mRST=180o. Find Arc Measures

23. Arc Addition Postulate • The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

24. b. a. = + A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures. mACD = mAC + mCD b. a. mACD mAB mBC mAC mAC Find Arc Measures EXAMPLE 2 = 29o + 108o = 137o = 137o + 83o = 220o

25. Examples 5 3 1 . TQ 6 4 2 . RST . TS . TQR . QRT . QS Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc. minor arc, 120° major arc, 240° semicircle, 180° minor arc, 160° minor arc, 80° semicircle, 180°

26. Congruent Circles • Two circles are congruent if they have the same radius. • Two arcs are congruent if they have the same measure and they are arcs of the same circle (or congruent circles). Are the red arcs congruent? Yes No

27. Theorem • In the same circle, two minor arcs are congruent if and only if their corresponding chords are congruent.

28. In the diagram, PQ, FGJK, and mJK= 80o. Find mFG So, mFG = mJK = 80o. EXAMPLE 1

29. Use the diagram of D. 1. If mAB= 110°, find mBC 2. If mAC = 150°, find mAB mBC = 110° mAB = 105° Try On Your Own GUIDED PRACTICE

30. Theorems • If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. • If one diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

31. Use the diagram of Eto find the length of AC. Diameter BDis perpendicular to AC. So, by the Theorem, BDbisects AC, and CF = AF. Therefore, AC= 2 AF =2(7) = 14. EXAMPLE 3

32. Try On Your Own Find the measure of the indicated arc in the diagram. 1. CD 2. DE mCD=mDE. mDE=72° 3. CE mCE=mDE + mCD mCE=72°+ 72° = 144° mCD=72°

33. In the diagram of C, QR = ST = 16. Find CU. Theorem • In the same circle, two chords are congruent if and only if they are equidistant from the center. CU = CV Use Theorem. 2x = 5x – 9 Substitute. Solve for x. x = 3 So, CU= 2x = 2(3) = 6.

34. Try On Your Own In the diagram, suppose ST = 32, and CU= CV = 12. Find the given length. 3. The radius of C QR= 32 UR= 16 The radius of C = 20 1. QR 2. UR

35. inscribed angle intercepted arc Measure of an Inscribed Angle Theorem • The measure of an inscribed angle is one half the measure of its intercepted arc. A ●C D B An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.

36. R S Q T mQTS = 2mQRS = 2 (90°) = 180° Example

37. Theorem • If two inscribed angles of a circle intercept the same arc, then the angles are congruent. E D F C

38. Try On Your Own mTV = 2m U = 2 38o = 76o. m G = mHF = (90o) = 45o ZYN ZXN ZXN 1 1 72° 2 2 Find the measure of the red arc or angle. 1. 2. 3.

39. Inscribed Polygons • A polygon is inscribed if all of its vertices lie on a circle. • Circle containing the vertices is a Circumscribed Circle.

40. Theorem A right triangle is inscribed in a circle if and only if the hypotenuse is a diameter of the circle. ●C

41. Theorem A quadrilateral is inscribed in a circle if and only if its opposite angles are supplementary. ●C y = 105° x = 100°

42. Try On Your Own Find the value of each variable. 1. 2. c = 62 x = 10 y = 112 x = 98

43. Theorem If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

44. = a. m1 12 (130o) (125o) 2 = b. m KJL Line mis tangent to the circle. Find the measure of the red angle or arc. = 250o = 65o

45. Try On Your Own = m XY m RST m1 12 (210o) (98o) (80o) 2 2 = = Find the indicated measure. = 105o = 196o = 160o

46. Angles Inside the Circle Theorem If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angles. (mBC + mDA) xo = 12 x°

47. The chords JLand KMintersect inside the circle. (mJM + mLK) xo = 12 12 xo (130o + 156o) = xo = 143 Find the value of x. Use Theorem 10.12. Substitute. Simplify.

48. Angles Outside the Circle Theorem • If a tangent and a secant,two tangents, or two secantsintersect outside a circle, then the measure of the angle formed is one half the differenceof the measures of the intercepted arcs. D

49. The tangent CDand the secant CBintersect outside the circle. (mAD – mBD) m BCD = 12 12 xo (178o – 76o) = x = 51 Find the value of x. Use Theorem 10.13. Substitute. Simplify.

50. Try On Your Own a y = 61o = 104o xo 253.7o Find the value of the variable. 5. 6.