Answer Key. 9.1 Parts of Circles. Chapter 9 Circles. CK12 Geometry Concepts 1. Answers. 1. diameter. 2. secant. 3. chord. 4.


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1 9.1 Parts of Circles 1. diameter 2. secant 3. chord 4. point of tangency 5. common external tangent 6. common internal tangent 7. the center 8. radius 9. chord 10. The diameter is the longest chord in any circle. 11. ʘB: 4 units, ʘC: 5 units, D: 2 units,.ʘe: ʘD ʘE because they have the same radius length. 13. They will have 2 common external tangents and no common internal tangents. 14. CE = Radius of ʘE + Radius of ʘC = = y = x 2 CK12 Geometry Concepts 1
2 9.2 Tangent Lines 1. Yes 2. No 3. Yes 4. x = x = x = 9 7. x = 3 8. x = 5 9. x = a) Yes, by AA. m CAE = m DBE = 90 and AEC BED by vertical angles. b) BC = 37 c) AD = 35 d) m C = Statement Reason 1. AB and CB with points of tangency at Given A and C. AD and DC are radii. 2. AD DC All radii are congruent. 3. DA AB and DC CB Tangent to a Circle Theorem 4. m BAD = 90 and m BCD = 90 Definition of perpendicular lines 5. Draw BD. Connecting two existing points 6. ADB and DCB are right triangles Definition of right triangles (Step 4) 7. DB DB Reflexive PoC 8. ABD CBD HL 9. AB CB CPCTC CK12 Geometry Concepts 2
3 12. a) ABCD is a kite. b) The line that connects the center and the external point B bisects ABC. 13. AT BT CT by the Two Tangents Theorem and the transitive property AW and WB both share point W and are perpendicular to VW because a tangent is perpendicular to the radius of the circle. Therefore A, B and W are collinear. CK12 Geometry Concepts 3
4 9.3 Arcs in Circles 1. minor arc 2. major arc 3. semicircle 4. major arc 5. minor arc 6. semicircle 7. Yes, CD DE 8. mcd mcae 228 because their corresponding central angles are congruent. 10. Yes, they are in the same circle with equal central angles. 11. Yes, the central angles are vertical angles and they are equal, which means the arcs equal. 12. No, we don t know the measure of the corresponding central angles x = x = x = ʘA ʘB CK12 Geometry Concepts 4
5 9.4 Chords in Circles x = 64, y = 4 8. x = 8, y = x 15.3, y x x = 9, y = x = x = x = x BF FD and BF 19. CA AF by Chord Theorem #4. FD by Chord Theorem # QS is a diameter by Chord Theorem # a) 120 b) 60 CK12 Geometry Concepts 5
6 9.5 Inscribed Angles in Circles 1. semicircle 2. congruent 3. chords 4. J L m JKL = 1 m JML 2 M K x = 180, y = x = 60, y = x = 30, y = CK12 Geometry Concepts 6
7 14. AC OD, so m CAB = m DOB by Corresponding Angles. A O C D B Also, m DOB = m DB and m CAB = 1 mcb, so 2 m DB = 1 mcb. This makes D the midpoint of CB Statement Reason 1. Inscribed ABC and diameter BD Given m ABE = x and m CBE = y 2. x + y = m ABC Angle Addition Postulate 3. AE EB and EB EC All radii are congruent 4. AEB and EBC are isosceles Definition of an isosceles triangle 5. m EAB = x and m ECB = y Isosceles Triangle Theorem 6. m AED = 2x and m CED = 2y Exterior Angle Theorem 7. mad = 2x and mdc = 2y The measure of an arc is the same as its central angle. 8. mad mdc mac Arc Addition Postulate 9. mac = 2x + 2y Substitution 10. mac = 2(x + y ) Distributive PoE 11. mac = 2m ABC Subsitution 12. m ABC = 1 2 m AC Division PoE CK12 Geometry Concepts 7
8 9.6 Inscribed Quadrilaterals in Circles 1. inscribed 2. opposite, supplementary x = 200, y = x = 68, y = x = 93, y = x = x = CK12 Geometry Concepts 8
9 9.7 Angles On and Inside a Circle 1. x = x = x = x = x = x = x = 60, y = x = x = x = x = Statement 1. Intersecting chords AC and BD. 2. Draw BC A D a B Given Construction Reason C 3. m DBC = 1 mdc 2 Inscribed Angle Theorem 4. m ACB = 1 2 mab Inscribed Angle Theorem 5. m a = m DBC + m ACB Exterior Angle Theorem 6. m a = 1 mdc + 1 mab 2 2 Substitution 13. inside, intercepted 14. on, half 15. No, by definition a tangent line cannot pass through a circle, so it can never intersect with any line inside of one. CK12 Geometry Concepts 9
10 9.8 Angles Outside a Circle 1. x = 63, y = x = x = x = x = x = 180, y = z = x = 180, y = x = x = x = x = 35, y = x = 47, y = x = 84, y = 156 CK12 Geometry Concepts 10
11 14. Statement 1. Intersecting secants AB and AC. Given 2. Draw. BE Construction B D A a Reason E C 3. m BEC = 1 mbc 2 4. m DBE = 1 mde 2 Inscribed Angle Theorem Inscribed Angle Theorem 5. m a + m DBE = m BEC Exterior Angle Theorem 6. m a = m BEC m DBE Subtraction PoE 7. m a = 1 mbc 1 mde Substitution m a = 1 (mbc mde ) Distributive Property a) b) c) CK12 Geometry Concepts 11
12 9.9 Segments from Chords 1. True 2. False 3. False 4. x = 5 5. x = 2 6. x = x = x = x = x = x = x = x = inches 15. Statement Reason 1. Intersecting chords AC and BE with Given segments a, b, c, and d. 2. AED BCD Theorem 98 EAD CBD 3. ADE~ BDC AA Similarity Postulate 4. a = d Corresponding parts of similar triangles c b are proportional 5. ab = cd Cross multiplication CK12 Geometry Concepts 12
13 9.10 Segments from Secants 1. x = 3 2. x = 6 3. x = x x = 6 6. Statement Reason 1. Secants PR and RT with segments a, given b, c, and d. 2. R R Reflexive PoC 3. QPS STQ Theorem RPS~ RTQ AA Similarity Postulate 5. c a c+d = a+b Corresponding parts of similar triangles are proportional 6. a(a + b) = c(c + d) Cross multiplication 7. z = 7 8. x = m = n = s = x = x = x x = 14 CK12 Geometry Concepts 13
14 9.11 Segments from Secants and Tangents 1. x = 5 2. x = 6 3. x = x = The error is in the set up. It should be10 10 = y (15 + y). The correct answer is y = x = x = x = x = x = 12, y = 3 CK12 Geometry Concepts 14
15 9.12 Circles in the Coordinate Plane 1. center: (5, 3), radius = 4 2. center: (0, 8), radius = 2 3. center: (7, 10), radius = center: (2, 0), radius = (x 4) 2 + (y + 2) 2 = (x + 1) 2 + (y 2) 2 = 7 7. (x 2) 2 + (y 2) 2 = 4 8. (x + 4) 2 + (y + 3) 2 = a) yes b) no c) no 10. (x 2) 2 + (y 3) 2 = (x 10) 2 + y 2 = (x + 3) 2 + (y 8) 2 = (x 6) 2 + (y + 6) 2 = a) (3.5, 5.5) b) 1, 1 c) y = x 2 d) y = x + 8 e) center: ( 5, 3) f) radius: 25 g) (x + 5) 2 + (y 3) 2 = (x 2) 2 + (y 2) 2 = (x + 3) 2 + (y 1) 2 = 289 CK12 Geometry Concepts 15
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