njctlcontent.njctl.org/courses/math/geometry-2015-16/circles/... · web view2015/04/08 · 20 10 =...

CirclesParts of a CircleClassworkUse the diagram of the circle with center A to answer the following:

1. Name the radii2. Name the chord(s)3. Name the diameter(s)4. If AC = 7, what does TC = ?5. If CT = 13, what does MA = ?6. Which is longer TC or MA? Justify.

7. Explain the difference between the radius of a circle and a chord.

Parts of a CircleHomeworkUse the diagram of the circle with center C to answer the following:

8. Name the radii9. Name the chord(s)10. Name the diameter(s)11. If CE = 8, what does BD = ?12. If BD = 19, what does CE = ?13. Which is longer DB or AB? Justify.

14. Explain the difference between the diameter of a circle and a chord.

Angles & ArcsClassworkIn C, ADis the diameter, m∠BCD=110°∧m∠ACE=80° . Find the measurement of each arc and classify the arc as a minor arc, major arc, or semicircle.

15. m AE16. m AB17. m ABD18. mEBD19. m BED20. m AED21. m ADB

Two concentric circles have center P, PS = 6 and SU = 4.22. Which is greater: mRS∨mTU ?23. Which is greater: the length of RS∨the lengthof TU ?24. ∠TPU=90 °, how long would chord TU be?

Geometry – Circles ~1~ NJCTL.org

Angles & ArcsHomeworkIn C, ADis the diameter, m∠BCD=130 °∧m∠ ACE=60 °. Find the measurement of each arc and classify the arc as a minor arc, major arc, or semicircle.

25. m AE26. m AB27. m ABD28. mEBD29. m BED30. m AED31. m ADB

Two concentric circles have center P, PS = 3 and SU = 3.32. Which is greater: mRS∨mTU ?33. Which is greater: the length of RS∨the lengthof TU ?34. ∠TPU=90 °, how long would chord TU be?

Arc Length & RadiansClassworkPARCC type QuestionsIn C, ADis the diameter, m∠BCD=110° ,m∠ ACE=80°, and CE = 5, find the following

35. lengthof AE36. lengthof m AB37. lengthof AD38. lengthof EBD39. lengthof BED40. lengthof ADE41. lengthof ADB42. If the central angle of a circle has measure 60o and makes a minor arc with length 15,

what is the radius?

43. If the arc of a circle has length 8π and the circumference of the circle is 24π, what is the measure of the central angle that intercepts the arc?

In #44-49, convert the degrees of the angle to radians, or the radians of the angle to degrees. Use 3.14 as your value of 𝜋 .

44. 20°45. 135°46. 343°47. 5 radians48. 3.5 radians

49.3π2 radians


Arc Length & RadiansHomeworkPARCC type QuestionsIn C, ADis the diameter, m∠BCD=130 ° ,m∠ACE=60°, and CE= 8, find the following

50. lengthof AE51. lengthof m AB52. lengthof AD53. lengthof EBD54. lengthof BED55. lengthof ADE56. lengthof ADB

57. If the central angle of a circle has measure 80o and makes a minor arc with length 12, what is the radius?

58. If the arc of a circle has length 10π and the circumference of the circle is 30π, what is the measure of the central angle that intercepts the arc?

In #59-64, convert the degrees of the angle to radians, or the radians of the angle to degrees. Use 3.14 as your value of 𝜋 .

59. 17°60. 150°61. 321°62. 4 radians63. 2.5 radians

64.π6 radians

Chords, Inscribed Angles & TrianglesClass WorkSolve for the variable in each problem. C is the center of the circle.65. 66. 67.

68. 69. 70.


71. 72. 73.

74. 75. 76.

77. 78. 79.

80. 81.

PARCC type Questions82. The figure to the right shows a circle with center H, diameter GF,

and inscribed ∆ FGJ . HF = 12. Let m∠GJF=(x+25)° and m∠ JGF=x°.a) Find the value of x.

Choose the correct option for each blank. Answer choices are givenin the boxes below each blank.

b) The length of JF is _______________ because __________________.

83. Point P is the center of a circle. RT is the diameter of the circle. Point U is a point on the circle, different from R and T.a) Determine if the following statements are always, sometimes, or never true.

1) RT > RU


12less than 12greater than 12

∆ JHF is equilateralm∠ JHF<60 °m∠ JHF>60 °

2) m∠TRU=12(m∠UPT )

3) m∠RTU=90 °4) m∠TRU=2(m∠RTU )

b) If m∠PUT=50°, what is m∠RPU?

Chords, Inscribed Angles & TrianglesHomeworkSolve for the variable in each problem. C is the center of the circle.84. 85. 86.

87. 88. 89.

90. 91. 92.

93. 94. 95.

96. 97. 98.

99. 100.


PARCC type Questions101. The figure to the right shows a circle with center C, diameter BD,

and inscribed ∆ BDE. BD = 28. Let m∠BED=(3x )° and m∠EBC=x°.a) Find the value of x.

Choose the correct option for each blank. Answer choices are givenIn the boxes below each blank.

b) The length of DE is ______________ because ______________.

102. Point M is the center of a circle. JK is the diameter of the circle. Point L is a point on the circle, different from J and K.a) Determine if the following statements are always, sometimes, or never true.

1) ML > KL

2) m∠KJL=12(m∠ JKL)

3) m∠KLJ=90 °4) LM=2(KJ )

b) If m∠ JKL=25 °, what is m∠ JML?

Tangents & SecantsClasswork

103. Draw a tangent line to the circle at M.104. What is the difference between a chord and a secant?

Draw the common tangents for each set of circles.105. 106. 107.

108. If a circle has a center of (7,6) and is tangent to the x-axis, how big is the radius?109. If a circle has a center of (7,6) and is tangent to the y-axis, how big is the diameter?

Solve for the variable in each problem. C is the center of the circle.


14less than 14greater than 14

∆ ECD is equilateralm∠ECD<60 °

110. 111. 112.

113. 114. 115.

116. 117. 118.

119. 120. 121.

122. 123. 124.

125. 126. 127.

PARCC type Question128. The figure shows two semicircles with centers K & M. The semicircles are tangent to each other at point J, and QN is tangent to both circles at N & O. If KL = JP = 12, what is OQ?


Tangents & SecantsHomework129. Draw a tangent line to the circle at A.130. What is the difference between a tangent and a secant?Draw the common tangents for each set of circles.131. 132. 133.

134. If a circle has a center of (3, -6) and is tangent to the x-axis, how long is the radius?135. If a circle has a center of (3, -6) and is tangent to the y-axis, how long is the diameter?

Solve for the variable in each problem. C is the center of the circle.136. 137. 138.

139. 140. 141.

142. 143. 144.

145. 146. 147.

148. 149. 150.


151. 152. 153.

PARCC type Question154. The figure shows two semicircles with centers R & S. The semicircles are tangent to each other at point P, and UW is tangent to both circles at V & W. If QR = PT = 18, what is WV?

Segments & CirclesClassworkFind the value of the variable. C is the center of the circle.155. 156. 157.

158. 159. 160.

161. 162. 163.

Segments & CirclesHomeworkFind the value of the variable. C is the center of the circle.164. 165. 166.


167. 168. 169.

170. 171. 172.


Multiple ChoiceFor questions 1-4, use the diagram at the right of ⊙F1. Name a secant of the circle

a. FA b. AC c. BE d. BC

2. BF = 7 and tangent BE = 9, what is AE?a. 5.656 b. 11.402 c. 4.402 d.2.402

3. m∠BCA=20° and BD = 8, what is the length of BC?a. 1.396 b. 2.793 c. 9.774 d. 19.548

4. m∠BCA=20°, what is the measurement of BA in radians?a. 0.35 radians b. 0.70 radians c. 1.40 radians d. 2,292.99 radians

5. If AB is a diameter and m AC=50 °, then what is m ABC?a. 50 ° b. 130 ° c. 230 ° d. 310 °

6. Find the value of a.a. 200b. 300c. 240d. 20

7. If an angle measures 3 radians, what is its measurement in degrees?a. 30°b. 85.94°c. 171.89°d. 343.77°

8. Find the value of b. a. 70b. 110c. 150d. 210

9. Find the value of c.a. 65b. 35c. 30d. not enough information

10. Find the value of d.a. 20b. 40c. 50d. 70


11. Find the value of e.a. 7.5b. 8c. 8.5d. 9

12. Find the value of f.a. 2b. 3c. 4d. 6

13. Find the value of g.a. 2b. 5.3c. 8d. 10

14. Find the value of x.a. 3b. 6.75c. 9d. 15

Point H is the center of a circle. EF is the diameter of the circle. Point G is a point on the circle, different from E and F.

15. EF > HEa. Alwaysb. Sometimesc. Never

16. m∠EFG=90°a. Alwaysb. Sometimesc. Never

17. FG=EGa. Alwaysb. Sometimesc. Never

18. m∠EHG=2(m∠EFG)a. Alwaysb. Sometimesc. Never

19. If m∠FEG=38 °, what is m∠GHF?a. 38°b. 52°c. 76°d. 104°


Extended Response1. S, T, U, and V are points of tangency of ⊙ A and ⊙B. TH = 4x + 8, SH = 6x + 4,

HU = x + 2y, and HV = 4x - 2y.a. Find the value of x. b. Find the value of y.c. If AB = 25 and UB (not drawn) = 5, what is the

length of AT (not drawn)?

2. In the diagram AB∥CD and CD is a diameter.a. If m AB=40 ° find the m BC .b. If AB = 12 and CD = 20, how far from the center is AB?c. Using the information from parts a) and b), how long is ACB?

3. A triangle is inscribed in a circle creating three arcs. Two of the arcs are 80° and 130°. a. Draw a diagram for the given information above.

b. Find the measurement of the missing arc.

c. Find the measurements of all of the inscribed angles and list the angles in order from greatest to least.

4. The figure shows two semicircles with centers D & F. The semicircles are tangent to each other at point C, and BH is tangent to both circles at G & H. DC = CA = 20.

a. Determine the lengths of the radii in eachcircle. Draw additional radii in the diagram.

b. Determine the length of AB.c. Determine the length of GB.

Geometry - Circles ~13~ NJCTL.org

Answer Key1. Segments AT, AM, AC2. Segments JH, TC3. Segment TC4. 145. 6.56. Segment TC is longer because the

diameter is twice the radius.7. The radius is the segment that has

one endpoint as the center of the circle and the other endpoint on the circle. A chord is a segment that has 2 endpoints on the circle.

8. Segments CD, CB, and CE9. Segments AB, DB10. Segment DB11. 1612. 9.513. Segment DB, diameter is longest

chord of a circle14. The diameter is the longest chord and

the only chord that passes through the center.

15. 80°; minor16. 70°; minor17. 180°; semicircle18. 260°; major19. 250°; major20. 180°; semicircle21. 290°; major22. They are equal23. TU is longer24. 10√225. 60°; minor26. 50°; minor27. 180°; semicircle28. 240°; major29. 230°; major30. 180°; semicircle31. 310°; major32. They are equal33. TU is longer34. 6√235. 6.9836. 6.1037. 15.738. 22.6939. 21.8240. 22.6941. 25.31

42. 45/π43. 120° 44. 0.35 radians45. 2.36 radians46. 5.98 radians47. 286.62°48. 200.64°49. 270°50. 8.3851. 6.9852. 25.1353. 33.5154. 32.1155. 41.8956. 43.2857. 8.5958. 120°59. 0.30 radians60. 2.62 radians61. 5.60 radians62. 229.30°63. 143.31°64. 30°65. X=466. 42 degrees67. 30 degrees68. X=369. X=870. 50 degrees71. X=572. 84 degrees73. X=14574. 20 degrees75. 140 degrees76. 90 degrees77. X=170 degrees78. X=20 degrees79. X=9580. X = 80 degrees81. x = 1282. a) x = 55°

b) greater than 12 m∠ JHF>60 °

83. a) 1) Always 2) Always 3) Never 4) Sometimesb) m∠RPU=100 °

84. v=4


85. b= 80 degrees86. n=220 degrees87. F=40 degrees88. R=9.8589. x=490. x=891. 5092. k=14093. d=8094. h=60 degrees95. g=5.6696. d=8097. e=3598. n=6099. f = 110100. x = 18.5101. a) x = 30

b) 14 ∆ ECD is equilateral

102. a) 1) Sometimes 2) Sometimes 3) Always 4) Never b) m∠ JML=50°

103. Tangent line touches the circle at M104. A chord has endpoints on the circle,

while a secant passes through.105. Four tangent lines. Two of the

tangent lines touch the outsides of the two circles, while the other two make a diagonal in the middle of the two circles.

106. Two tangent lines on the outsides of the two circles.

107. One tangent line at the bottom108. R=6109. D=14110. x=12111. x=9112. x=4113. c=41114. g=8115. x=2, y=6116. c=10117. x=7118. x=8119. a=35120. k=40121. x=130122. h=220123. f=80

124. g=60125. 65126. b=130127. m=120128. OQ = √1152=24√2=33.94129. Tangent line passes through A130. A tangent “touches” at one point,

while a secant touches at two points131. Two tangent lines on the outside.

Two more tangent lines making a diagonal through the middle.

132. One tangent line through the center of the two touching circles. Two more tangent lines, one at the top and one at the bottom.

133. No tangent lines134. R=6135. R=6136. f=9137. t=25138. 2.49139. g=7140. g=10141. x=3; y=2142. j=12143. r=11144. x=7145. d=80146. x=70/3147. x=220148. 40 degrees149. 140 degrees150. x=210151. a=30 degrees152. d=135153. d=60 degrees154. WU = √2592=36√2=50.91

VU = √648=18√2=25.46 WV = 36√2−18√2=18√2=25.46

155. n=6.4156. x=8157. x=4158. x=2159. x=3160. x=5.48161. x=6162. x=9163. x=4164. n=4165. r=5


166. h=2167. x=8168. y=1169. k=3.37170. v=4.47171. x=2.25172. a=1.66

Unit ReviewMultiple Choice

1. C2. C3. C4. B5. D6. A7. C8. C9. C10. A11. D12. C13. A14. A15. A16. C17. B18. A19. C

Extended Response1. (a) 2 (b) 1.5 (c) 32. (a) 110 (b) 8 (c) 55.8513. (a) Note: the letters used in the

diagram below can be any random letters chosen.

(b) m PO=150 °(c) m∠OQP=75 ° m∠OPQ=65 ° m∠POQ=40 °

4. (a)

(b)2010

=40+x10+x

21=40+x10+ x

20+2 x=40+ x2 x=20+xx=20=AB

(c) 102+GB2=302100+GB2=900GB2=800GB=20√2=28.28


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