11. Concentric Circles: Circles that lie in the same plane and have the same center.


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1 Circles Definitions KNOW THESE TERMS 1. Circle: The set of all coplanar points equidistant from a given point. 2. Sphere: The set of all points equidistant from a given point. 3. Radius of a circle: The distance from the center of a circle to a point on the circle. 4. Diameter of a circle: A chord that contains the center of a circle. 5. Chord: A segment whose endpoints lie on the circle. 6. Secant: A line that contains the chord of a circle. 7. Tangent: A line that intersects a circle in eactly one point. 8. Congruent Circles: Circles that have congruent radii. 9. Tangent Circles: Coplanar circles that intersect at eactly one point. 10. Common tangent: A line or segment that is tangent to two circles. 11. Concentric Circles: Circles that lie in the same plane and have the same center. 12. Arc: part of the circumference of a circle. 13. Minor Arc: Part of a circle that measure less than Major Arc: Part of circle that measures between 180 and Adjacent Arcs: Arcs of the same circle that have eactly one point in common. 16. Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. 17. Semicircle: Two arcs of a circle that are cut off by a diameter. 18. Central Angle: An angle with its verte at the center of the circle. 19. Inscribed Angle: An angle whose verte is on a circle and whose sides contain chords of the circle. 20. Intercepted Arc: consists of endpoints that lie on the sides of an inscribed angle and all of the points of the circle between them. 21. Sector of a circle: Region bounded by two radii of the circle and their intercepted arc. 22. Segment of a circle: region bounded by an arc and its chord. 1
2 Identify the following; 1. AB 2. DF 3. WG 4. HJ 5. point H 6. CB 7. point A 8. BC 9. segment WG 10. If the radius of a circle is 2 2 cm, what is the length of the diameter? 11. Find the center of a circle whose endpoints of a diameter are (5, 4) and ( 1, 8). 12. Find the length of the radius of the circle in #19. Matching: Use circle O to name the following parts. Put the CAPITAL letter of the correct answer on the line. None of the choices will be used more than once. 1] AC 2] CD 3] AD A] radius B] diameter C] inscribed angle B 4] AOB D] central angle 5] CAD E] chord A 6] ADC 7] AE 8] BC F] secant G] tangent H] minor arc E D O C 9] OA I] major arc 10] BDC J] semicircle 2
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4 II. Find the center and radius of each circle. 1. ( 1) 2 + (y + 3) 2 = 4 2 C(, ) r = 2. ( + 1) 2 + ( y 5) 2 = 4 C (, ) r = 3. ( + 1) 2 + ( y + 10) 2 = 25 9 C (, ) r = y 2 = 144 C (, ) r = 6. C (, ) r = 4. ( ) 2 + y 2 = 56 C (, ) r = 7. III. Write the equation of the circle described. C (, ) r = 8. Center: (1, 2); radius = 3 cm 9. Center (3, 4); radius = 5 units Center (2, 0); radius = 3 2 mm 11. Center (1, 3); radius = cm Center (0, 0); radius = 10 units 13. Center (1, 6); radius = 4 10 units 4
5 Eample: EF and FG are tangent to circle H. Find EF. Circle O is inscribed in DFE. EX = 15, FY = 12, and DZ = Find the perimeter of DFE. 4. What kind of is XZD? F 5. What kind of a is FOY? 6. If m ZXD is 70, m D = 7. If m ZFO is 30, m YFO = Z O Y 8. If m ZOF is 50, m YOF = D X E AC is tangent to circle P at A. 9. If BC = 13 and AC = 5, find AB. A C 10. If BC = 20 and AC = 16, find AB. 11. If BC = 12 and AC = 6, find PB. P Q 12. If BC = 6 2 and AC = 6, find AP. 13. If PB = 4 and AC = 6, find BC. 14. If BC = 13 and AC = 5, find AP. B 5
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8 1. Suppose a chord of a circle is 24 cm long and is 9 cm from the center of the circle. Find the length of the radius. 2. The radius of a circle is 15 cm. Find the length of a chord 12 cm from the center of the circle. 3. The diameter of a circle is 10 inches long and the length of a chord is 6 inches. Find the distance from the chord to the center of the circle. True/False 1. The bisector of one chord of a circle is also the bisector of only one other chord. 2. If a line contains the center of a circle and the midpoint of one of the chords of the circle and is to the other chord, then the two chords are. 3. The distance from the center of a circle to a chord is the length of the segment whose endpoints are the center of the circle and the midpoint of the chord. 4. The closer a chord is to the center of a circle, the shorter it is. 5. Find the length of a chord of a circle of radius 13cm if its distance from the center of the circle is 5 cm. 6. Find the length of the radius of a circle if a 14 inch chord is 24 inches from the center of the circle. 7. How far from the center of a circle of radius 10 mm is a chord of length 16 mm? 8. A chord of a circle is 10 inches long and 12 inches from the center of the circle. Find the length of the diameter of the circle. 9. A diameter of a circle is 20 cm and a chord is 16 cm long. Find the distance from the chord to the center of the circle. 8
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11 Arcs, Central Angles, and Chords 1) Central Angle = to its intercepted arc = 55 2) Inscribed Angle = 1 its intercepted arc 2 = 80 3) An inscribed angle in a semicircle =. = 4) Opposite angles of an inscribed quadrilateral are. = y = 88 y 91 5) Angle formed by two chords intersecting in the interior of a circle = arc 2 arc = 88 6) Angle formed by two secant segments in the eterior of the circle = outer arc 2 inner arc = ) Angles inscribed in the same arc 8) Congruent chords have congruent. or intercepting congruent arcs are y = y = = 11
12 I. On the center of the circle. II. On the circumference of the circle. E a) b) c) d) e) f) g) h) FIND THE MEASURE OF ABC or the VALUE OF X. 1) 2) 3) A 90º 200º A A C B C 80º B C B 12
13 4) 5) 6) 7) 8) 9) 8) 9) 10) III. Inside the circle E Find the measure of each angle. a. b. c. d. 13
14 IV. Outside the circle E Find the value of the missing angle. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) Eample B Find the measure of the following: p= q= a= b= arc ZYW= arc YWX= 14
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17 Chord Chord: When 2 chords intersect inside a circle, the product of the segments of one chord = the product of the segments of the other chord. c a d b a b = c d E = E. 2 E E. 4 E. 5 = = Secant Secant: When 2 secant segments are drawn from an eternal point, the product of one whole secant segment and its eternal segment = product of the other whole secant segment and its eternal segment. (whole secant)( eternal piece) = (whole secant)(eternal piece) E = = = 17
18 Secant Tangent: When a secant segment and a tangent segment are drawn to Circle from an eternal point, the product of the whole Secant segment and its et. piece = (tangent) 2. (whole secant)(et. piece) = (tangent) 2 E1. E Special segment lengths in Circles Solve for = = = 4) 5) 6) = = = 18
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