1. Chapter 10: Circles

2. 10.1 Circles and Circumference
Name a circle by the letter at the center of the circle
Diameter- segment that extends from one point on the circle to another point on the circle through the center point
Radius- segment that extends from one point on the circle to the center point
Chord- segment that extends from one point on the circle to another point on the circle
Diameter=2 x radius (d=2r)
Circumference: the distance around the circle
C=2πr or C= πd

3. Circle X
Diameter-
Radius-
Chord- A
chord B
diameter E C X
radius D

4. Name the circle
Name the radii
Identify a chord
Identify a diameter

8. Find the circumference of a circle to the nearest hundredth if its radius is 5.3 meters.
b. Find the diameter and the radius of a circle to the nearest hundredth if the circumference of the circle is 65.4 feet.

11. 10.2 Angles, Arcs and Chords
10.2
Semi-circle: half the circle (180 degrees)
Minor arc: less than 180 degrees
Name with two letters
Major arc: more than 180 degrees
Name with three letters
Minor arc = central angle
Arc length:

12. B
Minor arc
Minor arc
Minor arc = AB or BC
Semicircle = ABC or CDA
Major arc = ABD or CBD
AB + BC = 180 C
Central angle X A D
Semicircle

13. Find the value of x.

14. Find x and angle AZE

16. Find the measure of each minor arc.

20. 10.3 Arcs and Chords
If two chords are congruent, then their arcs are also congruent
In inscribed quadrilaterals, the opposite angles are supplementary
If a radius or diameter is perpendicular to a chord, it bisects the chord and its arc
If two chords are equidistant from the center of the circle, the chords are congruent

21. A
If FE=BC, then arc FE = arc BC
Quad. BCEF is an inscribed polygon – opposite angles are supplementary
angles B + E = 180 & angles F + C = 180
Diameter AD is perpendicular to chord EC – so chord EC and arc EC are bisected B F C E D

22. *You can use the pythagorean theorem to find the radius B
You will need to draw in the radius yourself E
*You can use the pythagorean
theorem to find the radius
when a chord is perpendicular
to a segment from the center
XE = XF so chord AB = chord CD
because they are
equidistant from the center A X C F D

27. In the circle below, diameter QS is 14 inches long and chord RT is 10 inches long. Find VU.

30. 10.4 Inscribed Angles
Inscribed angle: an angle inside the circle with sides that are chords and a vertex on the edge of the circle
Inscribed angle = ½ intercepted arc
An inscribed right angle, always intercepts a semicircle
If two or more inscribed angles intercept the same arc, they are congruent

31. A
Inscribed angles:
angle BAC, angle CAD, angle DAE, angle BAD, angle BAE, angle CAE X B
Ex: Angle DAE = ½ arc DE C E D

32. A
Inscribed angle BAC intercepts a semicircle- so angle BAC =90
Inscribed angles GDF and GEF both intercept arc GF, so the angles are congruent B C D F E G

33. A. Find mX.

34. Refer to the figure. Find the measure of angles 1, 2, 3 and 4.

35. ALGEBRA Find mR.

36. ALGEBRA Find mI.

37. ALGEBRA Find mB.

38. ALGEBRA Find mD.

39. The insignia shown is a quadrilateral inscribed in a circle
The insignia shown is a quadrilateral inscribed in a circle. Find mS and mT.

40. 10.5 Tangents
Tangent: a line that shares only one point with a circle and is perpendicular to the radius or diameter at that point.
Point of tangency: the point that a tangent shares with a circle
Two lines that are tangent to the same circle and meet at a point, are congruent from that point to the points of tangency

41. Lines AC and AF are tangent to circle X at points B and E respectively
-B and E are points of tangency
Radius XB is perpendicular to tangent AC at the point of tangency
AE and AB are congruent because they are tangent to the same circle from the same point X C F E B A

42. A. Copy the figure and draw the common tangents to determine how many there are. If no common tangent exists, choose no common tangent.

48. 10.6 Secants, Tangents, and Angle Measures
Secant and Tangent
Interior angle = ½ intercepted arc
Two Secants:
Interior angle = ½ (sum of intercepted arcs)
Two Secants
Exterior angle = ½ (far arc – close arc)
Two Tangents

49. C
2 Secants/chords:
Angle 1 = ½ (arc AD + arc CB)
Angle 2 = ½ (arc AC + arc DB) B 2 1 A D

50. Secant ED intersects tangent FC at a point of tangency (point F)
Angle 1 = ½ arc FE
Angle 2 = ½ (arc EA – arc FB) 1 F B A 2 D C

51. A. Find x.

52. B. Find x.

53. C. Find x.

54. A. Find mQPS.

55. B.

56. A.

57. B.

58. 10.7 Special Segments in a Circle
Two Chords
seg1 x seg2 = seg1 x seg2
Two Secants
outer segment x whole secant =
outer segment x whole secant
Secant and Tangent
outer segment x whole secant = tangent squared
*Add the segments to get the whole secant

59. E D A F
2 chords:
AO x OB = DO x OC
2 secants:
EF x EG = EH x EI H O C I B G

60. A D
Secant and Tangent:
AD x AB = AC x AC C B

61. A. Find x.

62. B. Find x.

63. A. Find x.

64. B. Find x.

65. Find x.

66. Find x.

67. LM is tangent to the circle. Find x. Round to the nearest tenth.

68. Find x. Assume that segments that appear to be tangent are tangent.