1. Chapter 10: Circles
Geometry H

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 2. Name the radii
3. Identify a chord 4. Identify a diameter

6. Circles can intersect at 2, 1 or no points.

9. 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.

12. 10.2 Measuring Angles and Arcs
Central angle: an angle with a vertex at the center of the circle.
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:

13. Sum of Central Angles = 360 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

14. Find the value of x.

15. Find x and angle AZE

17. Find the measure of each minor arc.

21. 10.3 Arcs and Chords
If two chords are congruent, then their arcs are also congruent
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
In inscribed quadrilaterals, the opposite angles are supplementary

22. 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

23. *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

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

32. Some more challenging Q’s
Find the distance from the center of a circle to a chord that is 80m long if the diameter of the circle is 82m long.
Two circles intersect and have a common chord that is 16cm long. The centers are 21cm apart. If the radius is 10cm, find the radius of the other circle.

33. 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
Intercepted arc = 2* inscribed angle
An inscribed right angle, always intercepts a semicircle
If two or more inscribed angles intercept the same arc, they are congruent
In inscribed quadrilaterals, the opposite angles are supplementary

34. Intercepted arc: has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle.

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

36. 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

37. A. Find mX.

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

39. ALGEBRA Find mR.

40. ALGEBRA Find mI.

41. ALGEBRA Find mB.

42. ALGEBRA Find mD.

43. 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.

44. More problems
A 10 by 24 rectangle is inscribed in a circle. Find the radius of the circle.
Find the exact area of a circle whose diameter is 15 inches.

45. 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
Common Tangent: line, ray or segment that is tangent to two circles in the same plane.

47. 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

53. More Walk Around Problems

54. Common Tangents

55. How to deal with common tangent problems
Draw an appropriate diagram
Draw a segment connecting their centers
Draw the radii to the points of contact
Through the center of the smaller circle, draw a line parallel to the common tangent
Extend any radius if necessary to obtain right triangles and rectangle
Use the Pythagorean theorem and properties of a rectangle

56. Example

57. 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

61. 2 Secants/chords:
Angle 1 = ½ (arc AD + arc CB)
Angle 2 = ½ (arc AC + arc DB) C B 2 1 A D

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

63. A. Find x.

64. B. Find x.

65. C. Find x.

66. A. Find mQPS.

67. B.

68. A.

69. B.

70. 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

71. 10.7 Special Segments in a Circle
Chord segments: when two chords intersect inside a circle and divide each other into two segments.
Secant segment: a segment of a secant line that has exactly one endpoint on the circle.
External secant segment: a secant segment that lies in the exterior of the circle.
Tangent segment: segment of a tangent with one endpoint on the circle

73. 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

74. A D
Secant and Tangent:
AD x AB = (AC)2 C B

75. A. Find x.

76. B. Find x.

77. A. Find x.

78. B. Find x.

79. Find x.

80. Find x.

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

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

83. 10.8 Equations of Circles