Page 1 Central Angles & Arc Measures


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1 Geometry/Trig Unit 8 ll bout ircles! Name: ate: Page 1 entral ngles & rc Measures Example 1: JK is a diameter of ircle. Name two examples for each: K Minor rc:, Major rc:, M Semicircle:, Name Pair of djacent rcs:, J L Example 2: G is the center of the circle. G ^ m = 75 m = m = 2. m = 3. m G = 4. m G = 5. m = 6. m G = 7. m G = 8. m = 9. m = 10. m = 11. m = 12. m = 13. m = 14. m = 15. mg = 16. m = 17. mg = 18. m = 19. mg = 20. mg =
2 Page 2 Problem Set 1 irections: Solve for each indicated variable or segment. is the center of each circle. 1. z refers to an 2. = 8 arc measure. z y x x = y = z = 3. = = = y x 130 m = 120 x z z y z refers to an arc measure. y and z refers to arc measures. x = y = z = x = y = z = 5. m = m 1 = m 2 = Textbook Practice: p. 341 E #113; p WE #16, 10, 11
3 Page 3 Example Tab Theorem 91 Sketch the diagram: Observations: ill in the Measurements: m PR onclusion (Theorem 91): If a line is tangent to a circle, then the line is to the radius drawn to the point of tangency. Example 1: Given ircle with a radius length of 5. is a point of tangency. = 5 3. Example 2: Given ircle with a radius length of 7. is a point of tangency. = 24. G ind: = m = m = Example 3: Given JK is a diameter and KL is a tangent. The radius of the circle is 8. J ind: = G = G = m = Is G the midpoint of? mg = UI UIZ EXMPLE Given ircle and is a point of tangency. = 8 = 17 K 45 L ind: JK = JL = KL = Example uia uiz: / 5
4 Page 4 Tab 1 orollary of Theorem 91 Sketch the diagram: Observations: ill in the Measurements: onclusion (orollary to Theorem 91): Segments that are tangent to a circle from a point are. Example 1: and are points of tangency. lassify by sides: m = 32 m = m = Example 2: and are points of tangency. x = ½x + 9 = = Example 3: Points J, K, L, and M are points of tangency. ind the perimeter of quadrilateral. J K J = 4, K = 7, L = 3, M = 5 K 4x + 2 UI UIZ 1 L Points K and J are points of Tangency. J KL = 7; KJ = 6; m LKJ = 70 M L Perimeter: uia uiz 1: / 4
5 Page 5 Problem Set 2 irections: ind the value of each indicated variable and measure. 1. is a tangent of ircle. 2. and are tangents to the circle. 8 3 = 2x + 2 = 4x + 8 = 7x + 2 = 5x + 1 G x = = = m = G = m = = x = = = 3. P,, R, and S are points of tangency. P 4. and are tangent segments. m = S 9 9 R ind the perimeter of uadrilateral. = m = Textbook Practice: p. 335 WE #15
6 Page 6 Tab 2 Theorem 94 Sketch the diagram: ill in Measurements: mg mh Observations: onclusion (Theorem 94): In the same circle or in congruent circles, congruent chords intercept arcs. Example 1: ind all angle and arc measurements. m = 40 m = m = m = 140 m = m = Example 2: ind all angle and arc measurements concerning circle. m = 86 m = m = lassify by sides: If m = 128, then m = UI UIZ 2 = 9 = 9 uia uiz 2: / 4 m = 131 m = 33
7 Page 7 Tab 3 Theorem 95 Sketch the diagram: ill in Measurements: mg mh Observations: onclusion (Theorem 95): The diameter that is perpendicular to a chord the chord and its intercepted arc. Example 1: ind all measurements. is the center of the circle. S RT = R 15 T M 17 S = M = MS = SP = P Example 2: ind all measurements. is the center of the circle. m = m = m = m = m = m = m = 220 HLLENGE: If = 10, find. UI UIZ 3 S ^ RP S is a diameter. RP = 18 mrs = 70 R T P uia uiz 3: / 6 S
8 Page 8 Tab 4 Theorem 96 Sketch the diagram: To measure the distance between a point and segment, you must measure the distance. ill in Measurements: E G mhg mk Observations: onclusion (Theorem 96): In the same circle or in congruent circles, chords are equally distant from the center. Example 1: ind all measurements. is the center of the circle. K JP = NM = J LM = LN = P M = K = N L M m NL = Given: KP ^ J; NM ^ L J = L = 3 KP = 8 E UI UIZ 4 E ^ G ^ G = 8 m = 106 = 3 E = 3 You will need to drawn in M, K, and N to complete this problem. uia uiz 4: / 3
9 Page 9 Problem Set 3 omplete each problem from the Written Exercises on page 347. raw and label a diagram for each problem
10 Page 10 Tab 5 Theorem 97 & orollaries 1 & 2 Sketch the diagram: Recall Inscribed ngle: _ m m m m Observations: _ onclusion (Theorem 97): The measure of an inscribed angle is equal to the measure of its intercepted arc. orollary 1: Inscribed ngles that intercept the same arc are. orollary 2: n angle inscribed inside of a semicircle is. Example 1: ind all measurements. G J m GJ = mhj = Example 2: ind all measurements. me = 102 m E= 109 mg = m E = H mgh = m E = mhg = E m = 129 m = Example 3: ind all measurements. is a diameter. Round all decimal answers to the nearest tenth. N UI UIZ 5 M P iagram 2 = 26, = 24, = m = m = iagram 1 mmp = 122; mmn = 40 is a diameter; = 6; = 10 uia uiz 5: / 7
11 Page 11 Tab 6 Theorem 97 orollary 3 Sketch the diagram (include the four angle measurements): Observations: onclusion (orollary 3): If a quadrilateral is inscribed in a circle, then its opposite angles are. Example 1: ind all measurements. Given: M m LMJ = 73 J m MJK = 88 mmj = 102 K L ind: m JKL = m KLM = mmjk = mjk = mmlk = mlmj = mlmk = Example 2: UI UIZ 6 R S T Given: uadrilateral is inscribed in ircle. = 8 and = 15. ind each measure: iameter Length = Radius Length = m RST = 102 m SRP = 65 P m = uia uiz 6: / 4
12 Page 12 Problem Set 4 irections: ind each indicated measure X is the center of the circle. G 75 X 120 mg = m = m G = N is a diameter J 100 K P 40 N M L m JMK = m JLK = 5. R 6. m NP = mnp = S T m SRT = m = m = Textbook Practice: p. 353 E #4, 5, 6, 9; p. 354 WE #14, 6
13 Page 13 Tab 7 Theorem 98 Sketch the diagram: Observations: onclusion (Theorem 98): mg m m mg The measure of an angle formed by a chord and a tangent is equal to the measure of the intercepted arc. Example 1: ind all indicated measurements. is a point of tangency. Example 2: ind all indicated measurements. is a point of tangency. m = 78 x 72 m = m = m = x = m = m = UI UIZ 7 N M K is a point of tangency. m MKL = 74 J K L uia uiz 7: / 3
14 Page 14 Tab 8 Theorem 910 RULE: ngle = ½(igger rc Smaller rc) ase 1 Two Secants ase 2 Two Tangents ase 3 Secant & Tangent m 1 = m 2 = m 3 = Example 1: is a point of tangency. Example 2: and are points of tangency. m = 20 m = 115 m = m = m = m = m = 116 m = m = UI UIZ 8 is a point of tangency. m = 120 m = 50 G H uia uiz 8: / 4
15 Page 15 Problem Set 5 omplete each problem from the lassroom Exercises on pages raw an label a diagram for each Textbook Practice: p. 353 E #7, 8; p. 354 WE #5, 7, 8, 9; p. 360 #1521
16 Page 16 nswers to Examples Example Tab Theorem 91 Example 1: = 10 m = 60 m = 30 Example 2: = 25, G = 7, G = 18 m = 73.7 mg = 73.7 Is G the midpoint of? No Example 3: JK = 16, KL = 16, JL = 16 2 Tab 1 orollary to Theorem 91 Example 1: lassify by sides: Isosceles m = 74 m = 74 Example 2: x = 2, = 10, = 10 Example 3: Perimeter = 38 Tab 2 Theorem 94 Example 1: m = 70 m = 70 m = 140 m = 80 Example 2: m = 86 m = 86 lassify by sides: Isosceles m = 232 Tab 3 Theorem 95 Example 1: RT = 30 M = 8 S = 17 MS = 9 SP = 34 Example 2: m = 140 m = 70 m = 70 m = 70 m = 140 m = 20 HLLENGE: = 18.8 Tab 4 Theorem 96 Example 1: JP = 4 NM = 8 LM = 4 LN = 4 M = 5 K = 5 m NL = 36.9 Tab 5 Theorem 97 & orollaries 1 & 2 Example 1: m GJ = 46 mhj = 88 mg = 71 mgh = 251 mhg = 289 Example 2: m E= 51 m E = 51 m E = 51 m = 129 m = 64.5 Example 3: = 26, = 24, = 10 m = 67.4 m = 22.6 Tab 6 Section 9.5 orollary 3 Example 1: m JKL = 107 mmlk = 176 m KLM = 92 mlmj = 214 mmjk = 184 mlmk = 296 mjk = 82 Example 2: iameter Length = 17 Radius Length = 17 2 m = 28.1 Tab 7 Theorem 98 Example 1: m = 156 m = 204 m = 102 Example 2: x = 108 m = 108 m = 54 Tab 8 Theorem 910 Example 1: m = 75 m = 170 m = 285 m = 245 Example 2: m = 244 m = 64
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