circles - center for teaching &...

Geometry – Circles ~1~ NJCTL.org

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 𝑇𝐶 or 𝑀𝐴? 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 𝐷𝐵 or 𝐴𝐵 ? Justify.

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

Angles & Arcs Classwork

In C, 𝐴𝐷 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. 𝑚𝐴��

16. 𝑚𝐴��

17. 𝑚𝐴𝐵��

18. 𝑚𝐸𝐵��

19. 𝑚𝐵𝐸��

20. 𝑚𝐴𝐸��

21. 𝑚𝐴𝐷�� Two concentric circles have center P, PS = 6 and SU = 4.

22. Which is greater: 𝑚𝑅�� 𝑜𝑟 𝑚𝑇��? 23. Which is greater: the length of 𝑅�� or the length of 𝑇��? 24. ∠𝑇𝑃𝑈 = 90°, how long would chord 𝑇𝑈 be?


Angles & Arcs Homework

In C, 𝐴𝐷 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. 𝑚𝐴��

26. 𝑚𝐴��

27. 𝑚𝐴𝐵��

28. 𝑚𝐸𝐵��

29. 𝑚𝐵𝐸��

30. 𝑚𝐴𝐸��

31. 𝑚𝐴𝐷�� Two concentric circles have center P, PS = 3 and SU = 3.

32. Which is greater: 𝑚𝑅�� 𝑜𝑟 𝑚𝑇��? 33. Which is greater: the length of 𝑅�� or the length of 𝑇��? 34. ∠𝑇𝑃𝑈 = 90°, how long would chord 𝑇𝑈 be?

Arc Length & Radians Classwork PARCC type Questions

In C, 𝐴𝐷 is the diameter, 𝑚∠𝐵𝐶𝐷 = 110°,𝑚∠𝐴𝐶𝐸 = 80°, and CE = 5, find the following

35. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴��

36. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑚𝐴��


38. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐸𝐵��

39. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐵𝐸��

40. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐷��

41. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐴𝐷�� 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, 𝐴𝐷 is the diameter, 𝑚∠𝐵𝐶𝐷 = 130°,𝑚∠𝐴𝐶𝐸 = 60°, and CE= 8, find the following


51. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑚𝐴��


53. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐸𝐵��

54. 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐵𝐸��



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. 68. 69. 70.


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 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 given in 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

2) 𝑚∠𝑇𝑅𝑈 = 1

2 (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 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 given In 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 = 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 𝑄𝑁 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

QM PKL 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. 148. 149. 150.


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 𝑈𝑊 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

US TR PQ

V


167. 168. 169. 170. 171. 172.


Multiple Choice For questions 1-4, use the diagram at the right of ⊙ 𝐹 1. 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 𝑚𝐴�� = 50°, then what is 𝑚ABC?

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. 3 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. 𝐸𝐹 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

Geometry - Circles ~13~ NJCTL.org

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 𝐴𝑇 (not drawn)?

2. In the diagram AB ∥ 𝐶𝐷 and CD is a diameter.

a. If 𝑚𝐴�� = 40° find the 𝑚𝐵��. 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 𝐴𝐶��?

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 each

circle. Draw additional radii in the diagram.

b. Determine the length of 𝐴𝐵 .

c. Determine the length of 𝐺𝐵 .

BF ADE C

G

H


Answer Key 1. Segments AT, AM, AC 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) 𝑚𝑃�� = 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

QO

x

20

10

10 1020B

F AD CE

G

H

circles - center for teaching &...

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