Angles & Arcs Classwork. Geometry Circles ~1~ NJCTL.org. 7. Explain the difference between the radius of a circle and a chord.

Circles

Parts of a Circle

Classwork

Use the diagram of the circle with center A to answer the following: 1. Name the radii

2. 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 𝑇𝐶̅̅̅̅ or 𝑀𝐴̅̅̅̅̅? Justify.

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

Parts of a Circle

Homework

Use the diagram of the circle with center C to answer the following: 8. Name the radii

9. 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 𝐷𝐵̅̅̅̅ or 𝐴𝐵̅̅̅̅? Justify.

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

Angles & Arcs

Classwork

In C, 𝐴𝐷̅̅̅̅ is 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. 𝑚𝐴𝐸̂ 16. 𝑚𝐴𝐵̂ 17. 𝑚𝐴𝐵𝐷̂ 18. 𝑚𝐸𝐵𝐷̂ 19. 𝑚𝐵𝐸𝐷̂ 20. 𝑚𝐴𝐸𝐷̂ 21. 𝑚𝐴𝐷𝐵̂

Two concentric circles have center P, PS = 6 and SU = 4. 22. Which is greater: 𝑚𝑅𝑆̂ 𝑜𝑟 𝑚𝑇𝑈̂ ?

23. Which is greater: the length of 𝑅𝑆̂ or the length of 𝑇𝑈̂ ? 24. ∠𝑇𝑃𝑈 = 90°, how long would chord 𝑇𝑈̅̅̅̅ be?

Angles & Arcs

Homework

In C, 𝐴𝐷̅̅̅̅ is 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. 𝑚𝐴𝐸̂ 26. 𝑚𝐴𝐵̂ 27. 𝑚𝐴𝐵𝐷̂ 28. 𝑚𝐸𝐵𝐷̂ 29. 𝑚𝐵𝐸𝐷̂ 30. 𝑚𝐴𝐸𝐷̂ 31. 𝑚𝐴𝐷𝐵̂

Two concentric circles have center P, PS = 3 and SU = 3. 32. Which is greater: 𝑚𝑅𝑆̂ 𝑜𝑟 𝑚𝑇𝑈̂ ?

33. Which is greater: the length of 𝑅𝑆̂ or the length of 𝑇𝑈̂ ? 34. ∠𝑇𝑃𝑈 = 90°, how long would chord 𝑇𝑈̅̅̅̅ be?

Arc Length & Radians

Classwork

PARCC type Questions

In C, 𝐴𝐷̅̅̅̅ is the diameter, 𝑚∠𝐵𝐶𝐷 = 110°, 𝑚∠𝐴𝐶𝐸 = 80°, and CE = 5, find the following 35. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐸̂ 36. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑚𝐴𝐵̂ 37. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐷̂ 38. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐸𝐵𝐷̂ 39. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐵𝐸𝐷̂ 40. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐷𝐸̂ 41. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐷𝐵̂

42. 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 radians 48. 3.5 radians 49. 3𝜋 2 radians

Arc Length & Radians

Homework

PARCC type Questions

In C, 𝐴𝐷̅̅̅̅ is the diameter, 𝑚∠𝐵𝐶𝐷 = 130°, 𝑚∠𝐴𝐶𝐸 = 60°, and CE= 8, find the following 50. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐸̂ 51. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑚𝐴𝐵̂ 52. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐷̂ 53. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐸𝐵𝐷̂ 54. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐵𝐸𝐷̂ 55. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐷𝐸̂ 56. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐷𝐵̂

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 radians 63. 2.5 radians 64. 𝜋 6 radians

Chords, Inscribed Angles & Triangles

Class Work

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

65. 66. 67.

71. 72. 73.

74. 75. 76.

77. 78. 79.

80. 81.

PARCC type Questions

82. The figure to the right shows a circle with center H, diameter 𝐺𝐹̅̅̅̅, 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 given in the boxes below each blank.

b) The length of 𝐽𝐹̅̅̅ is _______________ because __________________.

83. Point P is the center of a circle. 𝑅𝑇̅̅̅̅ 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 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? (10x - 2)° (5x + 2)° C 12 less than 12 greater than 12 ∆𝐽𝐻𝐹 is equilateral 𝑚∠𝐽𝐻𝐹 < 60° 𝑚∠𝐽𝐻𝐹 > 60° 12 H F G J

Chords, Inscribed Angles & Triangles

Homework

Solve 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. (3x - 8)° (7x + 3)° C

PARCC type Questions

101. The figure to the right shows a circle with center C, diameter 𝐵𝐷̅̅̅̅, 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 given In the boxes below each blank.

b) The length of 𝐷𝐸̅̅̅̅ is ______________ because ______________.

102. Point M is the center of a circle. 𝐽𝐾̅̅̅ 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 = 1 2 (𝑚∠𝐽𝐾𝐿) 3) 𝑚∠𝐾𝐿𝐽 = 90° 4) 𝐿𝑀 = 2(𝐾𝐽) b) If 𝑚∠𝐽𝐾𝐿 = 25°, what is 𝑚∠𝐽𝑀𝐿?

Tangents & Secants

Classwork

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.

110. 111. 112. 28 C B D E 14 less than 14 greater than 14 ∆𝐸𝐶𝐷 is equilateral 𝑚∠𝐸𝐶𝐷 < 60° 𝑚∠𝐸𝐶𝐷 > 60°

113. 114. 115.

116. 117. 118.

119. 120. 121.

122. 123. 124.

125. 126. 127.

PARCC type Question

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

Tangents & Secants

Homework

129. Draw a tangent line to the circle at A.

130. What is the difference between a tangent and a secant?

O Q M P K L J N

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.

151. 152. 153.

PARCC type Question

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

Segments & Circles

Classwork

Find the value of the variable. C is the center of the circle.

155. 156. 157.

158. 159. 160.

161. 162. 163.

Segments & Circles

Homework

Find the value of the variable. C is the center of the circle.

164. 165. 166.

W

U

S

T

R

P

Q

V

167. 168. 169.

Multiple Choice

a. FA̅̅̅̅ b. AC̅̅̅̅ c. 𝐵𝐸⃡⃗⃗⃗⃗ d. 𝐵𝐶⃡⃗⃗⃗⃗ 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 BĈ ?

a. 1.396 b. 2.793 c. 9.774 d. 19.548

4. 𝑚∠𝐵𝐶𝐴 = 20°, what is the measurement of BÂ in radians?

a. 0.35 radians b. 0.70 radians c. 1.40 radians d. 2,292.99 radians 5. If 𝐴𝐵̅̅̅̅ is a diameter and 𝑚𝐴𝐶̂ = 50°, then what is 𝑚ABĈ ?

a. 50° b. 130° c. 230° d. 310°

6. Find the value of a. a. 200

b. 300 c. 240 d. 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. 70 b. 110 c. 150 d. 210

9. Find the value of c. a. 65

b. 35 c. 30

d. not enough information 10. Find the value of d.

a. 20 b. 40 c. 50 d. 70 b° 70° 70° a° 160° 15° c° 80° Center

d°

40°

Center

11. Find the value of e. a. 7.5

b. 8 c. 8.5 d. 9

12. Find the value of f. a. 2

b. 3 c. 4 d. 6

13. Find the value of g. a. 2

b. 5. 3̅ c. 8 d. 10

14. Find the value of x. a. 3

b. 6.75 c. 9 d. 15

Point H is the center of a circle. 𝐸𝐹̅̅̅̅ is the diameter of the circle. Point G is a point on the circle, different from E and F. 15. EF > HE a. Always b. Sometimes c. Never 16. 𝑚∠𝐸𝐹𝐺 = 90° a. Always b. Sometimes c. Never 17. 𝐹𝐺 = 𝐸𝐺 a. Always b. Sometimes c. Never 18. 𝑚∠𝐸𝐻𝐺 = 2(𝑚∠𝐸𝐹𝐺) a. Always b. Sometimes c. Never 19. If 𝑚∠𝐹𝐸𝐺 = 38°, what is 𝑚∠𝐺𝐻𝐹? a. 38° b. 52° c. 76° d. 104° h 5x 3x + 6

Extended Response

1. 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 𝐴𝑇̅̅̅̅(not drawn)?

2. In the diagram AB̅̅̅̅ ∥ 𝐶𝐷̅̅̅̅ and CD̅̅̅̅ is a diameter. a. If 𝑚𝐴𝐵̂ = 40° find the 𝑚𝐵𝐶̂ .

b. If AB = 12 and CD = 20, how far from the center is AB̅̅̅̅? c. Using the information from parts a) and b), how long is 𝐴𝐶𝐵̂?

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 𝐵𝐻⃗⃗⃗⃗⃗⃗ is tangent to both circles at G & H. DC = CA = 20.

a. Determine the lengths of the radii in each circle. Draw additional radii in the diagram. b. Determine the length of 𝐴𝐵̅̅̅̅.

c. Determine the length of 𝐺𝐵̅̅̅̅.

B

F

A

D

E

C

G

H

Answer Key

2. Segments JH, TC 3. Segment TC 4. 14

5. 6.5

6. 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 CE 9. Segments AB, DB

10. Segment DB 11. 16

12. 9.5

13. Segment DB, diameter is longest chord of a circle

14. The diameter is the longest chord and the only chord that passes through the center. 15. 80°; minor 16. 70°; minor 17. 180°; semicircle 18. 260°; major 19. 250°; major 20. 180°; semicircle 21. 290°; major 22. They are equal 23. TU is longer 24. 10√2 25. 60°; minor 26. 50°; minor 27. 180°; semicircle 28. 240°; major 29. 230°; major 30. 180°; semicircle 31. 310°; major 32. They are equal 33. TU is longer 34. 6√2 35. 6.98 36. 6.10 37. 15.7 38. 22.69 39. 21.82 40. 22.69 41. 25.31 42. 45/π 43. 120° 44. 0.35 radians 45. 2.36 radians 46. 5.98 radians 47. 286.62° 48. 200.64° 49. 270° 50. 8.38 51. 6.98 52. 25.13 53. 33.51 54. 32.11 55. 41.89 56. 43.28 57. 8.59 58. 120° 59. 0.30 radians 60. 2.62 radians 61. 5.60 radians 62. 229.30° 63. 143.31° 64. 30° 65. X=4 66. 42 degrees 67. 30 degrees 68. X=3 69. X=8 70. 50 degrees 71. X=5 72. 84 degrees 73. X=145 74. 20 degrees 75. 140 degrees 76. 90 degrees 77. X=170 degrees 78. X=20 degrees 79. X=95 80. X = 80 degrees 81. x = 12 82. a) x = 55° b) greater than 12 𝑚∠𝐽𝐻𝐹 > 60° 83. a) 1) Always 2) Always 3) Never 4) Sometimes b) 𝑚∠𝑅𝑃𝑈 = 100° 84. v=4

85. b= 80 degrees 86. n=220 degrees 87. F=40 degrees 88. R=9.85 89. x=4 90. x=8 91. 50 92. k=140 93. d=80 94. h=60 degrees 95. g=5.66 96. d=80 97. e=35 98. n=60 99. f = 110 100. x = 18.5 101. 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 M 104. 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 bottom 108. R=6 109. D=14 110. x=12 111. x=9 112. x=4 113. c=41 114. g=8 115. x=2, y=6 116. c=10 117. x=7 118. x=8 119. a=35 120. k=40 121. x=130 122. h=220 123. f=80 124. g=60 125. 65 126. b=130 127. m=120 128. OQ = √1152 = 24√2 = 33.94 129. Tangent line passes through A 130. A tangent “touches” at one point,

while a secant touches at two points 131. 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 lines 134. R=6 135. R=6 136. f=9 137. t=25 138. 2.49 139. g=7 140. g=10 141. x=3; y=2 142. j=12 143. r=11 144. x=7 145. d=80 146. x=70/3 147. x=220 148. 40 degrees 149. 140 degrees 150. x=210 151. a=30 degrees 152. d=135 153. d=60 degrees 154. WU = √2592 = 36√2 = 50.91 VU = √648 = 18√2 = 25.46 WV = 36√2 − 18√2 = 18√2 = 25.46 155. n=6.4 156. x=8 157. x=4 158. x=2 159. x=3 160. x=5.48 161. x=6 162. x=9 163. x=4 164. n=4 165. r=5

166. h=2 167. x=8 168. y=1 169. k=3.37 170. v=4.47 171. x=2.25 172. a=1.66 Unit Review Multiple Choice 1. C 2. C 3. C 4. B 5. D 6. A 7. C 8. C 9. C 10. A 11. D 12. C 13. A 14. A 15. A 16. C 17. B 18. A 19. C Extended Response 1. (a) 2 (b) 1.5 (c) 3 2. (a) 110 (b) 8 (c) 55.851 3. (a) Note: the letters used in the

diagram below can be any random letters chosen. (b) 𝑚𝑃𝑂̂ = 150° (c) 𝑚∠𝑂𝑄𝑃 = 75° 𝑚∠𝑂𝑃𝑄 = 65° 𝑚∠𝑃𝑂𝑄 = 40° 4. (a) (b) 20 10= 40+𝑥 10+𝑥  2 1= 40+𝑥 10+𝑥 20 + 2𝑥 = 40 + 𝑥 2𝑥 = 20 + 𝑥 𝑥 = 20 = 𝐴𝐵 (c) 102+ 𝐺𝐵2= 302 100 + 𝐺𝐵2 = 900 𝐺𝐵2 = 800 𝐺𝐵 = 20√2 = 28.28 130° 80° P Q O x 20 10 10 10 20 F A B D C E G H