Understanding Circles: Properties, Relationships, and Key Definitions

1. Bell work What is a circle?

2. Bell work Answer • A circle is a set of all points in a plane that are equidistant from a given point, called the center of the circle

3. Unit 3 : Circles: 10.1 Line & Segment Relationships to Circles (Tangents to Circles) Objectives: Students will: 1. Identify Segments and lines related to circles. 2. Use Properties of a tangent to a circle

4. CHORD CENTER TANGENT LINE • DIAMETER ALSO A CHORD RADIUS SECANT Lines and Segments related to circles Exterior Point • • Interior Point Point of tangency

5. Lines and Segments related to circles Center of the circle CENTER • P Circle P

6. Lines and Segments related to circles Diameter – from one point on the circle passing through the center (2 times the radius) CENTER • DIAMETER ALSO A CHORD

7. Lines and Segments related to circles Radius– Segment from the center of the circle to a point on the circle (1/2 the diameter) CENTER • RADIUS (I) = 1/2 the Diameter

8. Lines and Segments related to circles Chord – a segment from one point on the circle to another point on the circle CHORD • DIAMETER ALSO A CHORD

9. Lines and Segments related to circles Secant – a line passing through two points on the circle • SECANT

10. Lines and Segments related to circles Tangent – is a line that intersects the circle at exactly one point TANGENT LINE • • Point of Tangency

11. Semicircles Center Diameter Radius 9. Tangent 10. Secant 11. Minor Arc 12. Major Arc Label Circle Parts 5. Exterior 6. Interior 7. Diameter 8. Chord

12. Q P k (p. 597)Theorem 10.1 If a line is tangent to a circle, then it is perpendicular( _|_ ) to the radius drawn to the point of tangency. If line k is tangent to circle Q at point P, Then line k is _|_ to Segment QP. Tangent line

13. 3 cm P Q • R • Example 1 Find the distance from Q to R, given that line m is tangent to the circle Q at Point P, PR = 4 cm and radius is 3 cm. 4 cm m

14. Example 1 answer Use the Pythagorean Theorem a² + b² = c² 3² + 4² = c² 9 + 16 = c² √25 = √c² c = 5

15. 9 in P Q • R • Example 2 Given that the radius (r) = 9 in, PR = 12, and QR = 16 in. Is the line m tangent to the circle? 12 in 16 in m

16. Example 2 answer No, it is not tangent. Use the Pythagorean Theorem a² + b² = c² 9² + 12² = 16² 81 + 144 = 256 225 = 256 Since they are not = then the triangle is not a right triangle and thus the radius is not perpendicular to the line m, therefore the line is not tangent to the circle.

17. Intersections of Circles No Points of Intersection • CONCENTRIC CIRCLES – Coplanar circles that share a common center point

18. Intersections of Circles One Point of Intersection The Circles are tangent to each other at the point • Internal Tangent • External Tangent Common Tangents

19. Intersections of Circles Two Points of Intersection • •

20. • • P • (p. 598) Theorem 10.3 If two segments from the same exterior point are tangent to a circle, then they are congruent. R • S __ __ If SR and ST are tangent to circle P, __ __ SR  ST T

21. • • P • Example 3 Segment SR and Segment ST are tangent to circle P at Points R and T. Find the value of x. 2x + 4 R • S 3x – 9 T

22. Example 3 Answer __ __ Since SR and ST are tangent to the circle, then the segments are , so 2x + 4 = 3x – 9 -2x -2x 4 = x – 9 + 9 + 9 13 = x

23. Unit 3 : Circles: 10.2 Arcs and Chords Objectives: Students will: 1. Use properties of arcs and chords to solve problems related to circles.

24. 25 ft x Bell work Find the value of radius, x, if the diameter of a circle is 25 ft.

25. Central Angle A • CENTER P P • 60º • B Arcs of Circles CENTRAL ANGLE – An angle with its vertex at the center of the circle

26. Central Angle A • CENTER P P • 60º • B Arcs of Circles Minor Arc AB and Major Arc ACB MINOR ARC AB • MAJOR ARC ACB C

27. Central Angle A • CENTER P P • 60º • B Arcs of Circles Themeasure of the Minor Arc AB = the measure of the Central Angle The measure of the Major Arc ACB = 360º - the measure of the Central Angle Measure of the MINOR ARC = the measure of the Central Angle AB = 60º The measure of the MAJOR ARC = 360 – the measure of the MINOR ARC ACB = 360º - 60º = 300º 300º • C

28. Semicircles Center Diameter Radius 9. Tangent 10. Secant 11. Minor Arc 12. Major Arc Label Circle Parts 5. Exterior 6. Interior 7. Diameter 8. Chord

29. Arcs of Circles Semicircle – an arc whose endpoints are also the endpoints of the diameter of the circle; Semicircle = 180º 180º • Semicircle

30. A • • C • 80º 170º • B Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs AB + BC = ABC 170º + 8 0º = 2 5 0º ARC ABC = 250º

31. Example 1 Find m XYZ and XZ X Y • • • 75º P 110° • Z

32. (p. 605) Theorem 10.4 In the same circle or in congruent circles two minor arcs are congruent iff their corresponding chords are congruent

33. Arc DE = 100º E D F G Arc FG = (3x +4)º Congruent Arcs and Chords Theorem Example 1: Given that Chords DE is congruent to Chord FG. Find the value of x.

34. E D Chord DE = 25 in Chord FG = (3x + 4) in F G Congruent Arcs and Chords Theorem Example 2: Given that Arc DE is congruent to Arc FG. Find the value of x.

35. (p. 605) Theorem 10.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chords and its arcs. Chord Diameter P Congruent Arcs • Congruent Segments

36. (p. 605) Theorem 10.6 If one chord is the perpendicular bisector of another chord then the first chord is the diameter Chord 2 Chord 1: _|_ bisector of Chord 2, Chord 1 = the diameter P • Diameter

37. (p. 606) Theorem 10.7 In the same circle or in congruent circles, two chords are congruent iff they are equidistant from the center. (Equidistant means same perpendicular distance) Q T V Chord TS  Chord QR __ __ iff PU  VU • R U P S Center P

38. Example Find the value of Chord QR, if TS = 20 inches and PV = PU = 8 inches Q T V 8 in • R U P 8 in S Center P

39. Unit 3 : Circles: 10.3 Arcs and Chords Objectives: Students will: 1. Use inscribed angles and properties of inscribed angles to solve problems related to circles

40. Bell work 1 A C Find the measure of Arc ABC, if Arc AB = 3x, Arc BC = (x + 80º), and __ __ AB BC AB = 3xº BC = ( x + 80º ) B

41. Bell work 2 You are standing at point X. Point X is 10 feet from the center of the circular water tank and 8 feet from point Y. Segment XY is tangent to the circle P at point Y. What is the radius, r, of the circular water tank? Y 8 ft • X r 10 ft • P

42. Inscribed Angle Intercepted Arc Inscribed Polygons Circumscribed Circles Words for Circles Are there any words/terms that you are unsure of?

43. INTERCEPTED ARC, AB A B Inscribed Angles Inscribed angle – is an angle whose vertex is on the circle and whose sides contain chords of the circle. INSCRIBED ANGLE Vertex on the circle

44. INTERCEPTED ARC, AB A B Intercepted Arc Intercepted Arc – is the arc that lies in the interior of the inscribed angle and has endpoints on the angle. INSCRIBED ANGLE Vertex on the circle

45. (p. 613) Theorem 10. 8 Measure of the Inscribed Angle The measure of an inscribed angle is equal half of the measure of its intercept arc. Central Angle m∕_ ABC = ½ m AC A • CENTER P P B • • C Inscribed angle

46. Example 1 The measure of the inscribed angle ABC = ½ the measure of the intercepted AC. Central Angle A • Measure of the INTERCEPTED ARC = the measure of the Central Angle AC = 60º B • 30º • 60º m∕_ ABC = ½ mAC = 30º • C

47. Example 2 Find the measure of the intercepted TU, if the inscribed angle R is a right angle. T • R • • U

48. Example 3 Find the measure of the inscribed angles Q , R ,and S, given that their common intercepted TU = 86º Q T • TU = 86º R • • S U

49. (p .614) Theorem 10.9 If two inscribed angles of a circle intercepted the same arc, then the angles are congruent Q T IF∕_ Qand∕_ Sboth intercepted TU,then ∕_ Q∕_ S • • • S U

50. Inscribed vs. Circumscribed Inscribed polygon – is when all of its vertices lie on the circle and the polygon is inside the circle. The Circle then is circumscribed about the polygon Circumscribed circle – lies on the outside of the inscribed polygon intersecting all the vertices of the polygon.