Pre-Test Name Date Use the following figure to answer Questions 1 through 6. A B C F G E D 1. What is the center of the circle? The center of the circle is point G. 2. Name a radius of the circle. A radius of the circle is GA (or GF or GC). 3. Name a chord of the circle that is not a diameter. A chord of the circle that is not a diameter is AC. 4. Name a diameter of the circle. A diameter of the circle is FC.. Name a secant of the circle. A secant of the circle is AB. 6. Name a tangent of the circle and identify its point of tangency. A tangent of the circle is ED, and its point of tangency is point E. Chapter Assessments 101
Pre-Test PAGE 2 Use the following figure to answer Questions 7 through 11. 24 A F E m AB 72 72 B C P D 48 m AF 24 m CD 48 7. What kind of angle is APB? What is its measure? Explain how you determined your answer. Angle APB is a central angle, because its vertex is at the center of a circle. Its measure is 72º, because the measure of a central angle is the same as the measure of the arc it intercepts. 8. What kind of angle is CED? What is its measure? Explain how you determined your answer. Angle CED is an inscribed angle, because its vertex is on the circle. Its measure is 24º, because the measure of an inscribed angle is half of the measure of the arc it intercepts. 9. What kind of arc is FBC? What is its measure? Explain how you determined your answer. Arc FBC is a semicircle. Its measure is 180º, because a semicircle is half of a circle. And, a circle has a measure of 360º. 10. What kind of arc is BC? What is its measure? Explain how you determined your answer. Arc BC is a minor arc, because it is less than a semicircle. Because m FA m AB m BC m FBC and FBC is a semicircle, 24º 72º m BC 180º. So, the measure of BC is 84º. 11. What kind of arc is BFC? What is its measure? Explain how you determined your answer. Arc BFC is a major arc, because it is more than a semicircle. Because m BC m BFC 360º, then 84º m BFC 360º. So, m BFC is 276º. 102 Chapter Assessments
Pre-Test PAGE 3 Name Date 12. Find the measure of LMN. Show all your work. P 42 L V M Q N 174 m/ LMN 1 2 (m LN m VP) 1 (174º 42º) 2 108º The measure of / LMN is 108º. 13. Find the measure of RST. Show all your work. R W U 3 V S 77 T m/ RST 1 2 (m RT m UV) 14. In circle A below, the line through point Y is tangent to the circle. Find the measure of XYZ. Show all your work. A 1 (77º 3º) 2 21º The measure of / RST is 21º. Y X Z 102 m/ XYZ 1 2 (m XY) 1 2 (102º) 1º The measure of / XYZ is 1º. Chapter Assessments 103
Pre-Test PAGE 4 Use the following figure to answer Questions 1 through 17. 4 mm D A G B Q E H F C m AC 94 AB 4 mm 1. What is the length of BC? Explain how you determined your answer. Because the diameter GH is perpendicular to the chord AC, it bisects that chord. So, AB > BC, and the length of BC is 4 millimeters. 16. What is the length of DF? Explain how you determined your answer. Because chords AC and DF are the same distance from the center of the circle, they are congruent. So, the length of DF is 8 millimeters. 17. What is the measure of DF? Explain how you determined your answer. Because chords and are congruent, their corresponding minor arcs are congruent. So, m AC DF DF is 94º. 18. In circle J below, the line through point K is tangent to the circle. What is the measure of JKL? Explain how you determined your answer. J L K Because KL is tangent to the circle, radius JK is perpendicular to the line at the point of tangency. So, m/ JKL is 90º. 104 Chapter Assessments
Pre-Test PAGE Name Date 19. Find the length of AB. Show your work and leave your answer in terms of. B A 100 9 cm O Arc length of AB : 100º 360º 2 (9) 18 18 The arc length of AB is centimeters. 20. Find the area of sector XYO. Show all your work and leave your answer in terms of. O 6 in. 4 X Y Area of sector XYO: The area of sector XYO is 4º 360º (62 ) 1 8 36 9 2 9 2 square inches. Chapter Assessments 10
Pre-Test PAGE 6 21. In Circle O below, OPQ has a height of 6 feet and a base length of 16 feet. Find the area of the segment of the circle. Show all your work. Use 3.14 for and round your answer to the nearest hundredth if necessary. P 10 ft O 140 Q Area of segment: 140º 360º (10 2 ) 1 2 (16)(6) 7 18 100 48 The area of the segment is about 74.11 square feet. 30 (3.14) 48 9 < 74.11 106 Chapter Assessments
Post-Test Name Date Use the following figure to answer Questions 1 through 6. J P K M L V Q N 1. What is the center of the circle? The center of the circle is point P. 2. Name a radius of the circle. A radius of the circle is PN (or PQ or PK). 3. Name a chord of the circle that is not a diameter. A chord of the circle that is not a diameter is MN (or JV). 4. Name a diameter of the circle. A diameter of the circle is KQ.. Name a secant of the circle. A secant of the circle is JV. 6. Name a tangent of the circle and identify its point of tangency. A tangent of the circle is and its point of tangency is point K. KL. Chapter Assessments 107
Post-Test PAGE 2 Use the following figure to answer Questions 7 through 11. W 37 64 V R X 96 S T U 7. What kind of angle is VXU? What is its measure? Explain how you determined your answer. Angle VXU is a central angle, because its vertex is at the center of the circle. Its measure is 96º, because the measure of a central angle is the same as the measure of the arc it intercepts. 8. What kind of angle is RSW? What is its measure? Explain how you determined your answer. Angle RSW is an inscribed angle, because its vertex is on the circle. Its measure is 32º, because the measure of an inscribed angle is half of the measure of the arc it intercepts. 9. What kind of arc is WRT? What is its measure? Explain how you determined your answer. Arc WRT is a semicircle. Its measure is 180º, because a semicircle is half of a circle. And, a circle has a measure of 360º. 10. What kind of arc is TU? What is its measure? Explain how you determined your answer. Arc TU is a minor arc, because it is less than a semicircle. Because m and is a semicircle, m TU m m TUW TU 96º 37º 180º. So, the measure of UV m VW TUW TU is 47º. 11. What kind of arc is TWU? What is its measure? Explain how you determined your answer. Arc TWU is a major arc, because it is more than a semicircle. Because 360º, then 47º m 360º. So, m m TU m TWU TWU TWU is 313º. 108 Chapter Assessments
Post-Test PAGE 3 Name Date 12. Find the measure of BFC. Show all your work. A E F 2 D B 3 C m/ BFC 1 2 (m BC m ED) 1 (3º 2º) 2 39º The measure of / BFC is 39º. 13. Find the measure of JLN. Show all your work. L M 42 K Q N J 110 m/ JLN 1 2 (m JN m KM) 1 (110º 42º) 2 34º The measure of / JLN is 34º. Chapter Assessments 109
Post-Test PAGE 4 14. In circle X below, the line through point B is tangent to the circle, and the measure of ADB is 236º. Find the measure of ABC. Show all your work. A D X 236 B C m/ ABC 1 2 (m ADB) 1 2 (236º) 118º The measure of / ABC is 118º. Use the following figure to answer Questions 1 through 17. R Y U 3 in. T W S V Z X m RW 88 WT 3 in. 1. What is the length of RT? Explain how you determined your answer. Because the diameter YZ is perpendicular to the chord RW, it bisects that chord. So, RT > TW, and the length of RT is 3 inches. 16. What is the length of SX? Explain how you determined your answer. Because chords RW and SX are the same distance from the center of the circle, they are congruent. So, the length of SX is 6 inches. 17. What is the measure of SX? Explain how you determined your answer. Because chords RW and SX are congruent, their corresponding minor arcs are congruent. So, m SX is 88º. 110 Chapter Assessments
Post-Test PAGE Name Date 18. In circle D below, the line through point E is tangent to the circle. What is the measure of DEF? Explain how you determined your answer. D E F Because EF is tangent to the circle, radius DE is perpendicular to the line at the point of tangency. So, m/ DEF is 90º. 19. Find the length of FG. Show your work and leave your answer in terms of. F Q 120 1 cm G Arc length of FG : 120º 360º 2 (1) 1 3 30 10 The arc length of FG is 10 centimeters. 20. Find the area of sector ABC. Show all your work and leave your answer in terms of. in. 36 B A C Area of sector ABC: The area of sector ABC is 36º 360º ( 2 ) 1 10 2 2 2 square inches. Chapter Assessments 111
Post-Test PAGE 6 21. In Circle P below, triangle PRS has a height of 9.2 feet and a base length of 1.4 feet. Find the area of the segment. Show all your work. Use 3.14 for and round your answer to the nearest hundredth if necessary. R 12 ft 80 P S Area of segment: 80º 360º (12 2 ) 1 2 (1.4)(9.2) 2 9 144 70.84 32(3.14) 70.84 < 29.64 The area of the segment is approximately 29.64 square feet. 112 Chapter Assessments
Mid-Chapter Test Name Date Use the following information and figure for Questions 1 through 9. Clive works for a company that photographs and documents crop circles. The following diagram represents one of the company s most recent findings. Point D is the center of the circle. One of Clive s responsibilities is to make an accurate diagram of each crop circle that is discovered. To save time and money, Clive measures only a few of the dimensions and then uses what he knows about circles to fill in the rest of the measurements on his diagram. 20 ft 30 A G B C D 20 ft F 60 E DF = 20 ft m AB 30 m FE 60 Find the measure of each of the angles, arcs, and segments named in Questions 1 through 9. Use complete sentences to explain how you found each of your answers. 1. FDE The measure of / FDE is 60º, because the measure of a central angle is equal to the arc it intercepts. 2. ADC The measure of / ADC is 60º, because it is vertical to / FDE. 3. AC The measure of AC is 60º, because a minor arc is equal to the measure of the central angle that intercepts it. Chapter Assessments 113
Mid-Chapter Test PAGE 2 4. AC The measure of AC is 20 feet, because if two minor arcs in a circle are congruent, then their corresponding chords are congruent.. FBE The measure of / FBE is 30º, because the measure of an inscribed angle is half the measure of the arc it intercepts. 6. AGB The measure of / AGB is 4º, because the measure of an angle formed by two intersecting chords is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 7. DE The measure of DE is 20 feet, because it is a radius of the circle. 8. AE The measure of AE is 40 feet, because it is a diameter of the circle. 9. ACE The measure of ACE is 180º, because it is a semicircle. 114 Chapter Assessments
Mid-Chapter Test PAGE 3 Name Date Use the following scenario and figure for Questions 10 through 13. The following is a diagram of another crop circle for which Clive must provide the appropriate measurements. Point K is the center of the circle. 2 Q I L MN is tangent to circle K at point M. H K M m QL 2 m LHM 274 87 J N m HJ 87 Find the measure of each of the angles and arcs named in Questions 10 through 13. Use complete sentences to explain how you found each of your answers. 10. HIJ 1 The measure of / HIJ is 31º, because it is equal to the difference of the measures of HJ and QL. 2 11. KMN The measure of / KMN is 90º, because a radius is perpendicular to a line that is tangent to a circle at the point of tangency. 12. LMN The measure of / LMN is 137º, because an inscribed angle is equal to half the measure of the arc it intercepts. 13. LM Because m LM + m LHM = 360º, then m LM + 274º = 360º. So, m LM = 86º. Chapter Assessments 11
Mid-Chapter Test PAGE 4 Use the following diagram and scenario for Questions 14 through 16. The following is a diagram of the final crop circle for which Clive must find measurements. P Q R K S T PS = 22 ft KM ~ = MX Z Y M N myv = 72 U X W V Find the measure of each of the angles, arcs, and segments named in Questions 14 through 16. Use complete sentences to explain how you found each of your answers. 14. RU The measure of RU is 22 feet, because chords that are the same distance from the center of the circle are congruent. 1. KMZ The measure of / PMZ is 90º, because if a diameter of a circle bisects a chord, then it is perpendicular to the chord. 16. KX Because KX and YV are the same distance from the center of the circle, they are congruent. The measure of KX is 72º, because if chords are congruent, then their intercepted arcs are congruent. 116 Chapter Assessments
End of Chapter Test Name Date Use the following diagram to find the measures of the angles, arcs, and segments named in Questions 1 through 10. Write your answers in the table provided and explain how you found your answers. 108 A 46 B C Q is the center of the circle. BC CD G H Q 11 cm GE 8 cm F E 46 D K Angle, arc, or segment Measure Explanation 1. BQC 108º The measure of a central angle is equal to the measure of the arc it intercepts. 2. BDC 4º The measure of an inscribed angle is one half of the measure of the arc it intercepts. 3. DC 108º If two chords in a circle are congruent, then their corresponding arcs are congruent 4. FDC. EAC 6. BC 7. GH 8. FG 180º 4º 11 cm 4 cm 26º The measure of a semicircle is 180º. The measure is equal to one half the difference between m EC and m GB. Congruent chords have the same measure. If a diameter is perpendicular to a chord, then the diameter bisects the chord. The measure of is 180º, because it is a semicircle. so m m FBC FG m GB m BC m FBC, FG 46º 108º 180º. Therefore, m FG 26º. 9. BDK 108º The measure of an inscribed angle is one half the measure of the arc it intercepts. 10. FC D 36º The measure of an inscribed angle is one half the measure of the arc it intercepts. Chapter Assessments 117
End of Chapter Test PAGE 2 Use the following diagram for Questions 11 through 13. L 20 ft Q is the center of the circle. Q 34 M LM is tangent to circle Q at point L. NM is tangent to circle Q at point N. N 11. What is the measure of NM? Use a complete sentence to explain how you found your answer. The measure of NM is 20 feet, because if two tangent segments to a circle are drawn from the same point outside the circle, then the tangent segments are congruent. 12. What is the measure of NLM? Use a complete sentence to explain how you found your answer. Triangle LMN is an isosceles triangle, which means that m/ NLM m/ LNM. Because m LMN 34º and m NLM m LNM m LMN 180º, it follows that 2(m NLM) 180º 34º 146º. So, m NLM 146º 2 73º. 13. What is the measure of QLN? Use a complete sentence to explain how you found your answer. The measure of / QLM is 90º, because a radius is perpendicular to a tangent segment at the point of tangency. Because m/ QLN m/ NLM m/ QLM and m NLM 73º, then m QLN 73º 90º. Therefore, m QLN 17º. 118 Chapter Assessments
End of Chapter Test PAGE 3 Name Date 14. Deborah would like to put edging along the circular edge of her flower garden. The following is a diagram of her flower garden. How much edging will she need if she just puts it along the circular part? Show all your work and use 3.14 for. Round your answer to the nearest hundredth if necessary. 9 8 ft Arc length: 9º 2 (8) 19 16 360º 72 < 13.26 Deborah will need about 13.26 feet of edging. 1. Jonathan has a circular pool in his backyard with an 18-foot diameter. He would like to pave a 6-foot-wide circle around his pool. How much paved area will Jonathan have around his pool? Show all your work and use 3.14 for. Area of swimming pool: A (9 2 ) 81 Area of outer circle: A (1 2 ) 22 Paved area around pool: A 22 81 < 144(3.14) 42.16 Jonathan will have about 42.16 square feet of paved area around his pool. Chapter Assessments 119
End of Chapter Test PAGE 4 16. A company has a circular card table with a 4-foot diameter. They want to remove a portion to provide a place for the dealer to stand. See the following diagram. How much surface area of the table will be left for those who are sitting at the table? Show all your work and use 3.14 for. Round your answer to the nearest hundredth if necessary. Portion to be removed for dealer 120 Area of whole table: Area of sector: A (2 2 ) 4 < 12.6 A 120º 360º (2 2 ) < 1 3 (4)(3.14) < 4.19 Area of table left for those sitting at the table: A 12.6 4.19 8.37 There will be about 8.37 square feet left for those sitting at the table. 17. Geneva has a circular table with a 6-foot diameter that she would like to put in her new kitchen. In order for it to fit up against the wall, she must cut off the portion of the table that is shaded in the following diagram. The measure of the central angle is 100º. How much surface area will she lose when she removes this part of the table? Show all your work and use 3.14 for. 4.6 ft 1.9 ft Area of sector: Area of triangle: A 100º 360º (3 2 ) 18 9 2 A 1 (4.6)(1.9) 4.37 2 Area of segment: A 4.37 2 < 7.8 4.37 3.48 Geneva will lose about 3.48 square feet of surface area of the table. 120 Chapter Assessments
Standardized Test Practice Name Date 1. A regular octagon is inscribed in a circle. x What is the measure of angle x in degrees? a. 4º b. 72º c. 90º d. 120º 2. In the circle below, Q is the center. Line YZ is tangent to the circle at point Y, and WY is a diameter. What is the measure of XYZ? Y Z 28 Q W X a. 28º b. 62º c. 90º d. 138º Chapter Assessments 121
Standardized Test Practice PAGE 2 3. An equilateral triangle is inscribed in a circle. B A C What is the measure of AB? a. 60º b. 90º c. 120º d. Cannot be determined 4. In the circle below, P is the center. What is the measure of KLQ? M 30 N 84 K L P Q a. 30º b. 3º c. º d. 70º 140 122 Chapter Assessments
Standardized Test Practice PAGE 3 Name Date. In the circle below, N is the center. The measure of P is 23º and the measure of ST is 108º. What is the measure of QR? P S Q 23 108 N T R a. 46º b. 62º c. 8º d. 108º 6. In the following figure, AB and CB are tangent to both circles. The measure of AB is 1 centimeters. What is the measure of CE? A D cm B E C a. centimeters b. 7 centimeters c. 9 centimeters d. 10 centimeters Chapter Assessments 123
Standardized Test Practice PAGE 4 7. In the following figure Q is the center and LM NP. What is the measure of LM? L P M Q 98 N a. 49º b. 98º c. 120º d. 138º 8. The following is a diagram of Jeanie s garden. She would like to put decorative fencing all the way around the edge of her garden. Approximately how much fencing will Jeanie need? 9 ft 140 a. 21 feet b. 22 feet c. 39 feet d. 40 feet 124 Chapter Assessments
Standardized Test Practice PAGE Name Date 9. An interior designer would like to make a pattern in the carpet in a client s square family room. The following is a diagram of the design. The width of the room is 20 feet. Approximately how much carpet will the designer need for the area that is represented by the shaded portion in the following diagram? 6 ft a. 0 square feet b. 113 square feet c. 200 square feet d. 201 square feet 10. What is the area of the shaded portion of the following circle? 6 mm 70 a. 7 square millimeters b. 29 square millimeters c. 36 square millimeters d. 43 square millimeters Chapter Assessments 12
Standardized Test Practice PAGE 6 11. What is the approximate area of the segment? 8 in. a. 32 square inches b. 16 square inches c. ( 16 + 32) square inches d. ( 16 32) square inches 12. In the following figure, diameter AB bisects chord XY. Which of the following statements must be true? A Y X B a. Diameter AB is congruent to chord XY. b. Diameter AB is parallel to chord XY. c. Diameter AB is longer than chord XY. d. Diameter AB is perpendicular to chord XY. 126 Chapter Assessments