Practice Exam IV - Chapters 8 & 9
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 . a. True b. False 2. For an isosceles triangle with length of leg s and length of base b, the perimeter P is given by P = 2s + b. a. True b. False 3. The area of the trapezoid with base lengths A = h( + ). a. True b. False
and
and altitude h is given by
4. The area of an equilateral triangle with sides of length s is given by A = a. True b. False
.
5. If the base of the pyramid shown is a square, then the pyramid is a regular square pyramid. a. True b. False 6. Prisms and pyramids are types of polyhedrons. a. True b. False 7. If two triangles have the same areas and the same length of altitude, these triangles must be congruent. a. True b. False 8. An octagonal prism has 9 faces. a. True b. False Copyright Cengage Learning. Powered by Cognero.
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Practice Exam IV - Chapters 8 & 9 9. For a square with apothem of length a, the perimeter equals 4a. a. True b. False 10. For a circle of radius r, the area is given by A = 2 r. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 11. If the area of a regular octagon is 25 octagon. a. 50 b. 25 c. 50
, find the area of the regular octagon whose sides are twice those of the first
d. None of These
12. For a triangle whose perimeter measures 36 units, the radius of the inscribed circle is 3.Find the area of the triangle. a. 27 b. 54 c. 108 d. None of These
13. A food container is in the shape of a right circular cylinder. To the nearest cubic inch, find the exact volume of the food container if the radius of the base measures 2.3 inches and the height of the container is 6.2 inches. a. 52 b. 76 c. 92 d. 103 14. For a right circular cylinder with radius of base r and length of altitude h, the formula for the lateral area L is: a. b. c. d. None of These 15. Find the lateral area for a regular hexagonal prism in which the base edges are 4 inches long and the altitude is h = 6 inches. a. 36 b. 72 c. 144 d. None of These 16. In kite HJKL, HJ = HL and JK = LK. If m a. b. c.
= 90°, HJ = 12, and JK = 5, find the length of diagonal
.
d.
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Practice Exam IV - Chapters 8 & 9
17. For the right triangular prism shown, the base is a right triangle with sides of lengths 3 in, 4 in, and 5 in. If the prism has a height of 6 inches, find its volume. a. 24 b. 36 c. 72 d. 360 18. Each lateral face of a regular hexagonal pyramid has lateral edges of length 10 inches and a base edge of 12 inches. What is the lateral area of the pyramid? a. 48 b. 288 c. 384
d.
19. To the nearest tenth of square inch, find the area of the regular pentagon whose apothem measures 4.2 inches and whose sides measure 6.2 inches. a. 64.2 b. 65.1 c. 97.6 d. 130.2
20. In ,m a. 1.5 cm c. 6 cm
= 45° and OA = 6 cm. Find b. 3 cm d. 12 cm
.
21. For the triangle with sides of lengths a, b, and c, find an expression for the semiperimeter s of the triangle. 22. In
,m
= 90°. What fraction represents the part of the circumference that is the length of
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?
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Practice Exam IV - Chapters 8 & 9
23. A right circular cone has a radius of 6 inches and an altitude of 8 inches. Use area of the cone. 24. and If the area of
in order to find the exact total
have the same length of base and the same length of altitude. is 25 , find the area of .
25. Write the formula for the area A of any quadrilateral that has perpendicular diagonals of lengths and . 26. A square has an area of 13
. What is the exact length of each side of the square?
27. An equilateral triangle has an inscribed circle with the length of radius 3 inches. Find the exact length of each side of the equilateral triangle..
28. For , radii and and arc bound a sector. If m correct to the nearest tenth of a square inch.
29. In the form
= 100° and PX = 5.6 inches, find the area of the sector
, verify Euler’s Formula for the pyramid shown.
30. For a square whose length of apothem is a, find an expression containing a that represents the area A of the square.
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Practice Exam IV - Chapters 8 & 9
31. A square is inscribed in a circle. If the area of the circle is 28.25 and the area of the square is 18 area of the four-part region that lies outside the square but inside the circle. 32. For regular pentagon ABCDE, diagonals 33. The area of a circle is 27.5 about this circle.
34. If m
and
are drawn to form
. Find m
, find the
.
. To the nearest tenth of a square inch, find the area of the square that is circumscribed
= 45° and OA = 4.3 cm, find l
correct to the nearest tenth of a centimeter.
35. For a triangle with sides of lengths 13 cm, 14 cm, and 15 cm, the altitude to the 14 cm side separates that side into segments of lengths 5 cm and 9 cm. For the solid of revolution determined by revolving the triangular region about the 14 cm side, find its volume.
36. In
, chord
is 12 inches long and m
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= 90°. Find the exact area of the segment bounded by
and
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Practice Exam IV - Chapters 8 & 9
37. For
, the radius has length r and m
bounded by diameter
and
= 180°. In terms of r, write an expression for the perimeter of the sector
.
38. Like the pyramid, the volume formula for a right circular cone can be expressed as formula for the volume of the cone.
. State another form of the
39. A right rectangular prism has a base that measures 6 in by 8 in. If the volume of the “box” is 360 of the prism.
, find the height
40. Given that 1 foot = 12 inches, what is the number of square inches in 1 square foot?
41. 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: 42. Consider a circle with diameter length d, radius length r, and circumference C. Given that formula for the circumference of a circle is given by .
43. Using the drawing provided and fact that the area of a parallelogram is given by triangle is given by
, explain why the
, show that the area of a
.
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Practice Exam IV - Chapters 8 & 9
44. Use the drawing provided to explain the following theorem. “The area A of a regular polygon whose apothem has length a and whose perimeter is P is given by
.”
Given: Regular polygon apothem so that
with center O and length s for each side;
Prove:
45. Where is the degree measure for the arc of a sector of a circle, the ratio of the area of the sector to that of the area of the circle is given by
. Use this ratio to explain why
the area of the sector is given by [Note: In the figure, the sector with arc measure
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. is bounded by radii
,
, and
.]
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Practice Exam IV - Chapters 8 & 9 Answer Key 1. True 2. True 3. False 4. False 5. False 6. True 7. False 8. False 9. False 10. False 11. d 12. b 13. d 14. b 15. c 16. d 17. b 18. b 19. b 20. a 21. s = (
)
22. 23. 24. 12.5 Copyright Cengage Learning. Powered by Cognero.
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Practice Exam IV - Chapters 8 & 9 25. A = 26.
inches
27.
inches
28. 27.4 29. 5 + 5 = 8 + 2 30. A = 4 31. 10.25 32. 36° 33. 35.0 34. 3.4 cm 35. 36. (18 - 36) 37. 2r + r or (2 + )r 38. 39. 7.5 in 40. 144 41. 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 . Similarly, , , and . Thus, the area of quadrilateral is one half of that of rectangle . But the area of
is
, so
is given by
.
42. Given that , we use the Multiplication Property of Equality to obtain Because the length of the diameter of a circle is twice that of a radius, . By substitution, or . Copyright Cengage Learning. Powered by Cognero.
.
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Practice Exam IV - Chapters 8 & 9 43. For the parallelogram ( Addition Postulate, triangles that have equal areas. By substitution,
) with base length b and altitude length h, the area is given by . By the Area. But diagonal of separates the parallelogram into 2 congruent and by division (or multiplication),
.
44. From center O, we draw radii , , , , and . Because the radii are congruent to each other and the sides of the regular polygon are all congruent to each other as well, , , , , and are all congruent to each other by SSS. Each of the congruent triangles has an altitude length of . Further, the length of each base of a triangle is s, the length of side of the polygon. Therefore, the area of the regular polygon is
Because the sum of the sides equals perimeter P, we have 45. Given that
, it follows that
of the circle is given by
. . Where r is the length of radius of the circle, the area
. By substitution, it follows that
.
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