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1 PRACTICE EXAM IV Sections 6.1, 6.2, Indicate whether the statement is true or false. 1. For a circle, the constant ratio of the circumference C to length of diameter d is represented by the number. 2. For a circle of radius r, the area is given by A = 2 r. 3. If, then the areas of and are equal. 4. The area of an equilateral triangle with sides of length s is given by A =. 5. When two secants intersect at a point in the exterior of a circle, the measure of the angle formed is one-half the positive difference of the measures of the two intercepted arcs. 6. The area of any regular polygon with length of apothem a and perimeter P is given by A = ap. 7. Because intercepts of, m = m. 8. In a circle (or congruent circles) containing two unequal chords, the shorter chord is nearer the center of the circle. 9. The area of the trapezoid with base lengths and and altitude h is given by A = h( + ). Copyright Cengage Learning. Powered by Cognero. Page 1

2 10. The region bounded by radii and and arc is known as a secant of the circle. Indicate the answer choice that best completes the statement or answers the question. 11. For cyclic quadrilateral ABCD, m and m. Determine the value of x. a. x = 50 b. x = 100 c. x = 120 d. x = The area of is Choosing a side of RSTV that measures 4.5 cm as the base, what is the length of the corresponding altitude? a. 7.8 cm b. 8.2 cm c cm d. None of These 13. If the area of a regular octagon is 25, find the area of the regular octagon whose sides are twice those of the first octagon. a. 50 b. 25 c. 50 d. None of These 14. Find the area of a square whose apothem has length 7 cm. a. 14 b. 28 c. 98 d In, and are minor arcs for which m m. Which must be true? a. b. m m c. is nearer to center O than d. None of These 16. In, m = 138. Find m. a. 64 b. 69 c. 138 d. None of These Copyright Cengage Learning. Powered by Cognero. Page 2

3 17. In, and in,. If AB = XY and CD = ZW, then: a. b. c. and have the same perimeter. d. and have the same area. 18. In, m = 45 and OA = 6 cm. Find. a. 1.5 cm b. 3 cm c. 6 cm d. 12 cm 19. In, m = 60 and BC = 12. Find AB. a. 3 b. c. 6 d. 20. In, and. How are and related? a. congruent b. same perimeters c. same areas d. similar 21. In, the length of radius is 6 inches. If m = 90 and m = 60, how much longer is chord than chord? Copyright Cengage Learning. Powered by Cognero. Page 3

4 22. A triangle has a perimeter of 40 and area of 60. Using A = rp, find the length of radius r for the inscribed circle for this triangle. 23. For the right triangle with sides of lengths 8, 15, and 17, find the length of the radius of the inscribed circle. 24. A square has an area of 13. What is the exact length of each side of the square? 25. In, m = 58. Find m. 26. Assuming that and in, what line segment is the altitude of this parallelogram with respect to side? 27. Find the measure of the angle formed by the hands of a clock at exactly 4:10 PM. 28. In, m = 90. What fraction represents the part of the circumference that is the length of? 29. For a regular octagon, the length of the apothem is a = 8.2 cm while the length of each side is s = 6.7 cm. To the nearest tenth of a square centimeter, find the area of the regular octagon. 30. and have the same length of base and the same length of altitude. If the area of is 25, find the area of. 31. For a square whose length of apothem is a, find an expression containing a that represents the area A of the square. Copyright Cengage Learning. Powered by Cognero. Page 4

5 32. To determine the measure of, which expression should be calculated. (m m ) or (m m )? 33. In this order, points R, S, T, U, and V are equally spaced on a circle in such a way. What type of figure is formed by the chords,,,, and? 34. For a circle, and are tangents from an external point D. How are and related? 35. If the area of quadrilateral ABCD is 27 and the area of is 11, find the area of pentagon ABXCD. 36. Explain why the following must be true. Given: Points A, B, and C lie on in such a way that ; also, chords,, and (no drawing provided) Prove: must be an isosceles triangle. Copyright Cengage Learning. Powered by Cognero. Page 5

6 37. Supply missing statements and missing reasons for the proof of the following theorem. An angle inscribed in a semicircle is a right angle. Given: with diameter and (as shown) Prove: is a right angle. S1. R1. S2. R2. S3. R3. The measure of a semicircle is 180. S4. or R4. S5. R Supply missing statements and missing reasons for the following proof. Given: Chords,,, and as shown Prove: S1. R1. S2. R2. S3. R3. If 2 inscribed intercept the same arc, these are. S4. R4. Copyright Cengage Learning. Powered by Cognero. Page 6

7 39. Use the drawing provided to explain why the following theorem is true. The tangent segments to a circle from an external point are congruent. Given: and are tangent to Prove: [Hint: Use auxiliary line segment.] 40. Use the drawing provided to explain the following theorem. The area of any quadrilateral with perpendicular diagonals of lengths and is given by. Given: Quadrilateral with at point F; and Prove: Copyright Cengage Learning. Powered by Cognero. Page 7

8 Answer Key 1. True 2. False 3. True 4. False 5. True 6. True 7. False 8. False 9. False 10. False 11. b 12. a 13. d 14. d 15. d 16. b 17. d 18. a 19. d 20. d 21. inches units 24. inches Copyright Cengage Learning. Powered by Cognero. Page 8

9 A = (m m ) 33. a regular pentagram Given that in, it follows that. But congruent arcs have congruent chords so that. Then must be an isosceles triangle. 37. S1. with diameter and (as shown) R1. Given R2. The measure of an inscribed angle is on-half the degree measure of its intercepted arc. S3. R4. Substitution Property of Equality S5. is a right angle. R5. Definition of a right angle. 38. S1. Chords,,, and as shown R1. Given R2. Vertical angles are congruent. S3. (or ) S4. R4. AA 39. Draw. Now and because the measure of an angle formed by a tangent and chord at the point of contact is one-half the measure of the intercepted arc. Then by substitution, so. Then because these sides lie opposite the congruent angles of. 40. To box the quadrilateral, we draw auxiliary lines as follows: through point D, we draw ; through point B, we draw ; through point A, we draw ; and through point C, we draw. The quadrilateral formed is a parallelogram that can be shown to have a right angle; this follows from the fact that is a parallelogram that contains a right angle at vertex F... so the opposite angle (at vertex R) must also be a right angle. Because is a diagonal of (actually rectangle, ; that is, a diagonal of a parallelogram separates the parallelogram into 2 congruent. Copyright Cengage Learning. Powered by Cognero. Page 9

10 Similarly,,, and. Thus, the area of quadrilateral is one half of that of rectangle. But the area of is, so is given by. Copyright Cengage Learning. Powered by Cognero. Page 10

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