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Test Bank For College Geometry: A Problem Solving Approach with Applications, 2nd Edition

Enhance your problem-solving skills with Test Bank For College Geometry: A Problem Solving Approach with Applications, 2nd Edition—your essential study companion.

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    GEOMETRY Chapter One True–False. Mark as true any statement that is always true. Mark as false any statement that is never true or that is not necessarily true. Be able to justify your answers. 1. An “exercise” can be solved by simply applying a routine procedure, but a “problem” is not routine and requires a well-thought-out plan of attack. 2. If there are infinitely many numbers involved in a problem, the Guess and Test problem-solving strategy is indicated. 3. If 12 coins are arranged in the shape of an equilateral triangle, then there are 5 coins along each side. 4. If the fourth phase of problem solving, “Looking Back,” we should look for other ways to solve the problem even if we got the ‘right’ answer. 5. When you are asked to make a generalization, you should use the Guess and Test problem-solving strategy. 6. If a problem involves a large array or diagram, a good strategy to try is “Use a Variable.” 7. If a problem involves a physical situation, the “Draw a Picture” strategy may be appropriate. 8. If a problem involves a sequence of numbers or figures, the “Draw a Picture” strategy may be appropriate. 9. If a problem asks you to make a prediction or generalization, the “Look for a Pattern” strategy may be appropriate. 10. If a problem asks “in how many ways” the “Look for a Pattern” strategy may be appropriate. 11. The fourth term in the sequence 3, 5, 7, . . . must be 9. 12. If a large cube is made up of 64 smaller cubes, then it has 16 cubes showing on each face of the larger cube. 13. Patterns in data may be easier to see if we arrange the data in tables. Multiple Choice. Mark the letter of the single BEST response. Be sure to read all the choices for each problem before deciding. 14. George Polya presented a four step process for problem solving which is still used extensively today. Which of the following is NOT a step in his basic process? (a) Devise a plan. (b) Understand the problem. (c) Carry out your plan. (d) Draw picture. (e) Look back to analyze your solution. 15. The Guess and Test problem-solving strategy can be useful when (a) there are only a few possible solutions to a problem. (b) the problem suggests an equation. (c) you are trying to develop a formula. (d) you need to find the measures of the angles in a triangle. (e) there is an unknown quantity related to known quantities

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    16. You may wish to use the “Solve a Simpler Problem” strategy if: (a) The problem involves very large or very small numbers. (b) There are a large number of cases. (c) A direct solution is too complex. (d) Both (a) and (b) are correct. (e) Both (a) and (c) are correct. 17. When using the Guess and Test Problem Solving Strategy you will probably be most successful if you (a) simply try possibilities at random. (b) use a means of organizing your guesses. (c) try to learn from each of the trials. (d) apply both parts (a) and (c). (e) apply both parts (b) and (c). 18. If a problem asks “how many cubes are in the nth figure,” an approach to solving the problem might be: (a) Guess and Test. (b) Draw a Picture. (c) Make a Table. (d) Look for a Pattern. (e) Use a Variable. 19. If a problem asks you to “Find a formula . . . “ you might begin by (a) drawing a sketch of the problem situation. (b) understanding the terms used in the problem. (c) looking for relationships between known quantities and the unknown quantity. (d) making a tables of values, using n as the unknown quantity. (e) trying any one of the above. 20. If you are given a sequence of numbers and asked to predict the next number in the sequence, a problem solving strategy you might reasonably try is (a) Look for a Pattern. (b) Draw a Picture. (c) Guess and Test. (d) Trial and Error. (e) None of these is a particularly good strategy. 21. If a problem begins with “There are 200 people at a dance . . .”, an approach to solving the problem may be (a) Draw a Picture. (b) Guess and Test. (c) Solve a Simpler Problem. (d) Deductive reasoning. (e) None of these is a good approach. 22. When we look at specific examples, observe a pattern, and draw a general conclusion, we are using (a) the Draw a Picture problem-solving strategy. (b) Deductive reasoning. (c) the Use a Variable problem-solving strategy. (d) The Guess and Test problem-solving strategy. (e) Inductive reasoning. 2 Chapter 1

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    23. Deductive reasoning requires us to (a) accept some initial conditions as true. (b) make a table. (c) look for a pattern. (d) solve a simpler problem. (e) draw a picture. 24. When using formal problem-solving strategies, we (a) must choose the right one for the problem. (b) should not mix them in one problem. (c) should always start by choosing a variable. (d) are using deductive reasoning. (e) None of these answers is always true. Fill in the Blanks. Complete each statement with a word or phrase that makes it true. 25. When solving a problem with geometric figures involved, we should use the problem-solving strategy. 26. When solving a problem in which you can easily present information, we should use the problem solving strategy. 27. If you wish to gain a better understanding of a complex problem, you might use the problem solving strategy. 28. The sum of the measures of the angles in a triangle is . 29. When a problem asks us to predict a sequence of numbers or figures, we should use the problem-solving strategy. 30. If we start with some general statements accepted as true and then draw conclusions from them, we are using reasoning. 31. If we are using reasoning, our answer may not be the only correct one. Writing. Write your answers concisely and completely. Feel free to use figures and/or tables to illustrate the points you are making. 32. List the four steps in the problem-solving process first introduced by George Polya and employed in all the problem-solving strategies. 33. Read and solve the following problem. Then write a paragraph describing the method you used to get your solution. Which problem-solving strategy(s) did you use and why? At a family dinner Grandmother, Aunt Suzie, Mary, Father and Richard are to sit along one side of the table. Grandmother must sit on one end. Aunt Suzie and Richard cannot be seated next to each other. Aunt Suzie wants to sit next to Mary. Mary must sit next to Grandmother. How should their seating arrangements be made? 34. Read and solve the following problem. Then write a paragraph describing the method you used to get your solution. Which problem-solving strategy(s) did you use and why? Toothpicks can be arranged to form rows of rectangles as shown. How many toothpicks are required to form a row with 30 rectangles in it? Chapter 1 3

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    35. Compare and contrast inductive and deductive reasoning. 36. Read and solve the following problem. Then write a paragraph describing the method you used to get your solutions. Which problem solving strategies did you use and why? The heptagonal numbers are the whole numbers that represent the following shapes. Find the number of dots in the nth heptagonal number. 37. Read and solve the following problem. Then write a paragraph describing the method you used to get your solution. Which problem-solving strategy(s) did you use and why? How many triangles of all sizes are there in the figure below? Exercises and Problems. 38. A large cube is formed by arranging 27 smaller white cubes of the same size into a cube shape. If three faces which share one corner of the larger cube are painted red, how many of the smaller cubes will have exactly two red faces? 39. If all the diagonals are drawn in a polygon with five sides (a pentagon), what is the maximum number of intersections of two diagonals? 40. The measure of one angle of a triangle is twice the measure of a second angle. The measure of the third angle is twelve degrees more than the measure of the first angle. What is the measure of the largest angle of the triangle? 41. Samuel, Martha, James, Katie and Rafael all go to the movies. James insists on sitting on the aisle. Samuel and Martha must sit next to each other. Rafael sits next to Samuel and next to James. Who is sitting next to Katie? 42. How many squares of all sizes are there in a 4 by 4 square? Sketch all of the squares on the provided 4 by 4 squares and briefly explain your work. Heptagonal Number 1 2 3 4 n? 4 Chapter 1

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    43. Shade exactly ten squares so that no three consecutive shaded squares line up vertically, horizontally or diagonally. 44. Toothpicks can be arranged to form triangles as shown below. How many toothpicks are required to form a triangle with ten toothpicks on a side? 45. Use inductive reasoning to determine the next figure in the sequence below. 46. Use inductive reasoning to find the 6th, 10th and nth terms in the following sequence: 0, 3, 8, 15, 24, . . . 47. How many squares will be in the tenth figure in the following sequence? 48. What fraction of the large triangle is shaded? 49. How can 24 cubes be stacked so that a minimum number of sides are exposed? How can 24 cubes be stacked so that a maximum number of sides are exposed? A side is one face of a smaller cube. Explain your thinking. 50. Determine the missing numbers in each of the following Fibonacci-type sequences (a) 1, 4, 5, 9, ,, , (b) 2, , 6, , 16, (c) 3, , , , 27 (d) 3, 7, 10, 17, 27 Chapter 1 5

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    51. Use inductive reasoning to write the 6th, 7th and 12th terms of each sequence. Then write a formula for the nth term. Briefly explain your method. (a) 3, 6, 9, 12, 15, . . (b) 4, 7, 11, 16, 22, . . . (c) , , , , , . . . Applications. 52. A landscape designer is working on the plans for a residential yard. Her clients want a privacy hedge planted on three sides of the rectangular backyard. (The side bordering the house will not be planted.) The plant selected for the hedge spans 5  when mature. If the backyard is 50 feet deep and 80 feet long and there is to be a plant at the corners of the hedge, how many hedge plants are required? (See sketch.) 53. A water system must be installed in a field as shown below. If the pipe comes in both 8-foot and 15-foot lengths, and cannot be cut, how many pipes of each length will be required? 54. Four areas are newly designated as parks as shown below. Hiking trails are to be constructed so that each park is directly connected with each of the others. What is the least number of trails needed? A B C D 92  38  69  38  80  50  Yard House 5 243 4 81 3 27 2 9 1 3 6 Chapter 1

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    55. A triangular raised bed of annual flowers is to be built next to the door of a new restaurant, as shown. If the angle of the flower bed near the door is 90  and the other angle by the walkway is three times the third angle, what are the measures of the angles of this flower bed? 56. A state map has county lines as shown. If the counties are to be colored and no two counties which share a border can have the same color, what is the minimum number of colors required for this map? Explain your solution. 57. There is to be a board meeting in the Success Unlimited Inc. boardroom. The chair of the board always sits at the head of the long rectangular table. There are three seats along each side of the table and one seat at the far end which no one ever uses. Six people should attend the meeting in addition to the chair. However, two people are out of town on assignment and a third person is ill. In how many different ways can the remaining people sit at the table? 58. A rancher wishes to fence a rectangular area of 500 sq. ft. He wants to use only whole-number dimensions. What dimensions will require the least amount of fencing? [Remember the area of a rectangle is A  l  w and the perimeter is P  2(l  w).] 59. Ten people are meeting at a conference. Each person is instructed to introduce herself or himself to each of the other participants at the conference. If it is estimated that each introduction should take 1.5 minutes, how much time should be allowed in the schedule for this activity? 60. A waste water system is being installed in a small town. The large pipes required to reach from the town to the water treatment plant come in 20-foot and 24-foot lengths. The city planner finds that twelve fewer of the 24-foot pipes will be required than if the 20-foot pipes were used. What is the distance from the town to the water treatment plant? 61. A school logo is made up of concentric circles as shown in the figure below. All the school colors, blue, green and yellow, are to be used. No two bordering regions can be the same color. If the center region must be yellow, how many different color schemes are possible? Restaurant Flower Bed Walkway Chapter 1 7

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    62. A mail carrier must deliver mail to houses in a subdivision shown in the following map. Lines represent streets and letters represent houses. List the houses in the order the mail carrier will deliver mail as she travels each street exactly once and mark the map to show the route. Can she start and stop at the same house? Explain. 63. These three shapes represent the first three figures in a sequence of figures made from small cubes. (a) How many cubes are in the 5th figure? Explain (b) Describe the number of cubes in the nth figure. Don’t simplify your answer, just list the pattern of the sum, for example, to sum 1 through n, you would write 1  2  . . .  (n  1)  n 64. The following is a 4  8 array of dots. (a) If 10 dots are removed, how can you arrange the remaining dots so that each row and each column of the array contain an odd number of dots? Sketch the new arrangement or explain why it cannot be done. (b) If 12 dots are removed, how can you arrange the remaining dots so that there is the same number of dots along each edge? Sketch the new arrangement or explain why it cannot be done. O L A B C D I J H G F E K N M 8 Chapter 1

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    65. Suppose three center cubes are removed from a large 3  3 cube and the resulting figure is dipped in red paint. How many cubes have red paint on 1 face? 2 faces? 3 faces? 4 faces? 5 faces? 6 faces? Explain. 66. Suppose all seven of the cubes from the center each face to the opposite face of a large 3  3 cube are removed and the resulting figure is dipped in red paint. How many cubes have red paint on 1 face? 2 faces? 3 faces? 4 faces? 5 faces? 6 faces? Explain. 67. Lynn purchased 24 decorative stone square pavers to create a border to enclose a rectangular brick patio. Each square paver measures 1 foot by 1 foot and the smaller rectangular bricks can be cut to fit any paver border configuration. What paver border configuration encloses the most brick patio area and what is that area? Explain. One border arrangement of pavers is shown here: Chapter 1 9

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    GEOMETRY Chapter Two True–False. Mark as true any statement that is always true. Mark as false any statement that is never true or that is not necessarily true. Be able to justify your answers. 1. If triangle ABC and triangle DEF are both 30  –60  triangles, then must be congruent to . 2. An angle of degree measure 180  is a reflex angle. 3. Postulates are statements we assume to be true without proof. 4. Any three points are contained in exactly one plane. 5. A right triangle has two acute angles. 6. An obtuse triangle has at least two obtuse angles. 7. All regular quadrilaterals have four lines of symmetry. 8. All regular polygons tessellate the plane. 9. The parallel sides in a trapezoid are called the legs of the trapezoid. 10. Regular polygons are polygons with all sides congruent and all angles congruent. 11. A soccer ball is a regular polyhedron. 12. The lateral faces of a rectangular prism meet at the apex of the prism. 13. The two opposite faces of a cone are called the bases of the cone and they can be oval (elliptically) shaped as well as truly circularly shaped. 14. A pyramid with rectangular faces is called a rectangular pyramid. 15. A sphere is a polyhedron. 16. The bases of a right regular prism may be congruent, parallel hexagons. 17. The lateral faces of a right square pyramid are isosceles triangles. 18. If there are 1.057 quarts in one liter and four quarts in one gallon, then there are 4.228 liters in one gallon. Multiple Choice. Mark the letter of the single BEST response. Be sure to read all the choices for each problem before deciding. 19. The measure of an acute angle is a number, n, such that: (a) 0   n  90  (b) 0   n  90  (c) 0   n  90  (d) 0   n  90  (e) 0   n  180  20. Which of the following are simple closed curves? (a) A circle. (b) A polygon. (c) A cube. (d) Both (a) and (b) are correct. (e) Both (a) and (c) are correct. DE AB

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    21. Two segments are congruent if (a) they intersect. (b) they are parallel. (c) they have the same length. (d) they share one endpoint. (e) None of these are correct. 22. If a triangle has two congruent sides and an angle with measure 90  , then it is called (a) right. (b) scalene. (c) isosceles. (d) acute. (e) Both (a) and (c) are correct. 23. If all the angles in a quadrilateral are right angles, then the quadrilateral might be (a) a trapezoid. (b) a kite. (c) a rectangle. (d) a square. (e) Both (c) and (d) are correct. 24. Assume the pentagon in the star is regular. The measure of  A is? (a) 36  (b) 54  (c) 72  (d) 108  (e) 144  25. An equilateral triangle has how many lines of symmetry? (a) 0 (b) 1 (c) 2 (d) 3 (e) 4 26. If a set of polygons forms a tessellation in the plane, then (a) they cover the plane with no gaps. (b) they are all congruent polygons. (c) none of the regions overlap. (d) Both (a) and (c) are correct. (e) Both (a) and (b) are correct. 27. The measure of a vertex angle in a regular polygon with nine sides is (a) 140  (b) 220  (c) 1260  (d) 20  (e) 40  28. Two faces of a prism intersect in (a) a vertex. (b) an edge. (c) a base. (d) a side. (e) Both (a) and (b) are correct. Chapter 2 11

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    29. The slant height of a right regular pyramid is (a) the perpendicular distance from its apex to the base of the pyramid. (b) the height of its lateral face. (c) the length of a base edge. (d) the length of a lateral edge. (e) None of these is correct. 30. Which of the following are regular polyhedra? (a) An octahedron. (b) A soccer ball. (c) A right circular cylinder. (d) A sphere. (e) Both (a) and (b) are correct. 31. If there are 2.54 cm in one inch then, in one foot there are (a) 30.48 cm. (b) 304.8 mm. (c) 0.3048 meters. (d) Both (a) and (b) are correct. (e) Answers (a), (b) and (c) are all correct. Fill in the Blanks. Complete each statement with a word or phrase that makes it true. 32. A line segment containing the center and with endpoints on a circle is called a . 33. A portion of a line that has one endpoint and extends indefinitely in one direction is called a(n) . 34. If the sum of the measures of two angles is 90  , then they are called . 35. The side of a right triangle that is opposite the right angle is called the . 36. If the sum of the vertex angles of a polygon is 540  , then it is called a(n) . 37. The faces of a right hexagonal prism are . 38. Name the five Platonic Solids. For each, give the name of the shape of the face and list the number of faces. a. Solid: Shape of faces: Number of faces: b. Solid: Shape of faces: Number of faces: c. Solid: Shape of faces: Number of faces: d. Solid: Shape of faces: Number of faces: e. Solid: Shape of faces: Number of faces: 39. For an oblique regular hexagonal pyramid, the name of the base is: and the name of the lateral faces is: Writing. Write your answers concisely and completely. Feel free to use figures and/or tables to illustrate the points you are making. 40. Compare and contrast the following types of quadrilaterals: trapezoid, parallelogram, rhombus, rectangle and square. 41. How are sides and angle measures used to classify quadrilaterals? 42. Consider a regular polygon with n sides. Discuss its characteristics, including sides, angles and symmetries. 12 Chapter 2

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    43. Which polygons can be used to make a regular tessellation? Explain why. Exercises and Problems. 44. In the figure,  FAB  30  ,  CAB  66  ,  GAD  23  , and G and F are collinear. (a) What is the measure of  DAE ? (b) Name three pairs of supplementary angles in this figure. (c) Name all of the pairs of complimentary angles in this figure. (d) Name two adjacent angles in this figure. 45. Convert each of the following angle measure to degrees and minutes. Round to the nearest minute if necessary. (a) 43.7  (b) 113.67  (c) 56.4  (d) 29.11  46. How many different rays have B as their initial point? Name them. 47. Points A, B, C and D are located on line l . Their corresponding coordinates are: A   2.35, B   .4, C  .7 and D  3.25. Find each of the following distances: (a) AC (b) BD (c) BC (d) AD 48. The measure of  X is 12  more than three times  Y . If  X and  Y are supplementary; find the measure of  X and classify it as acute, right, obtuse or straight. 49. On a standard clock, find the measure of the obtuse angle formed by the hour hand and the minute hand at 3:37 and 30 seconds. 50. In the figure, the measure of  APB is 72  and he measure of  DPE is 55  . What is the measure of  BPD? A P E D C B A B C D A B C D E A G D E B C F EA  BA Chapter 2 13

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    51. Given on the square lattice. (a) How many triangles can be drawn that have as a side? (b) How many of these triangles are isosceles triangles? (c) How many of these triangles are acute triangles? (d) How many of these triangles are right triangles? (e) How many of these triangles are obtuse triangles? 52. Several triangles are shown below with congruent sides, congruent angles and right angles shown. (a) Which ones are scalene triangles? (b) Which ones are isosceles triangles? (c) Which ones are right triangles? (d) Which ones are equilateral triangles? (e) Which ones are obtuse triangles? 53. The measures of two angles of a triangle are 38  and 102  .What is the measure of the third angle? 54. Use a protractor and a ruler to draw a quadrilateral with two angles of 120  and 3 congruent sides. Name the shape that you have created. 55. Use a protractor and a ruler to draw a hexagon where all of the vertex angles are 120  but the hexagon is not a regular hexagon. 56. Use this diagram of a regular pentagon to explain how to determine the formula for finding the vertex angle in a regular polygon. A G H M N P R Q T I B E D J L K F C A B AB AB 14 Chapter 2

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    57. In the following figure, each polygon is regular. Find the angle measures X and Y. 58. If the vertex angle of a regular polygon has measure 168  , how many sides does it have? 59. Given the pyramid below, verify Euler’s Formula: F  V  E  2. 60. Answer the following questions about the given prism. (a) Name the bases of the prism. (b) Name the lateral faces of the prism. (c) Name the faces that are hidden from view. (d) Name the edges of the prism. (e) Name the prism by type. Be as specific as possible. 61. When the following net is folded, what is the name of the resulting polyhedron? Give as complete a description as possible. 62. Use dimensional analysis to convert 0.3 days to minutes. 63. Use dimensional analysis to convert 246 ft/sec to meters per minute. (Remember that one inch is 2.54 centimeters.) C E F A D B X Y Chapter 2 15

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    64. Georgi ran a mile in 4 minutes and 18 seconds. (a) How fast did she run in miles per hour? (b) How fast did she run in miles per minute? (c) How fast did she run in kilometers per hour? (Remember there are 0.6214 miles in one kilometer) Applications. 65. A surveyor measures an angle to be 13  15  36  . In order to use this measure in his calculations he must convert it to degrees. Express this angle measure in degrees and a decimal fraction of one degree. 66. The bearing of a line segment is the acute angle that the line segment makes with a north- south line. If the bearing of is N 70  E and the bearing of is S 32  W, what is the measure of the obtuse angle;  QPR? 67. An airplane takes off from an airport at a bearing along , of N 13  E, and a second plane leaves the same airport at bearing N 30  W along . A third plane leaves the airport at S 87  W along . What is the measure of the angle between the first and second planes? Between the first and third planes? (Remember: The bearing of a line segment is the acute angle that the line segment makes with a north-south line.) 68. In order to complete an edge in a tiling job, a builder needs an isosceles trapezoid with a base that fits along the side of the regular hexagonal tiles he is using. Can he form the trapezoid in one cut? If so, how might he do this? If not, why not? 69. Using the given triangle lattice, if possible, sketch the following shapes. If not possible, explain why. (a) Sketch four non-congruent non-right parallelograms with S AB as one side. (b) Sketch all of the squares with S AB as one side. (c) Sketch all of the non-square rectangles with S AB as one side. (d) Sketch an isosceles trapezoid with S AB as one side. 70. For the given figure the hexagon is regular and the shaded regions are rhombi. (a) Describe the rotational symmetries of the figure or explain why none exist. (b) Sketch the lines of symmetries on the figure or explain why none exist. A A B AD AC AB PR PQ 16 Chapter 2

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    71. For the given figure both triangles are equilateral triangles. (a) Describe the rotational symmetries of the figure or explain why none exist. (b) Sketch the lines of symmetries on the figure or explain why none exist. 72. In the following figure: S BF and S AG are parallel and DEF is an isosceles triangle. (a) List two supplementary angles. (b) What shape is ABDEGHA? (c) What is the measure of  DFG ? (d) What is the measure of  BAH ? (e) How many obtuse interior angles are in the figure? List the angle names and measures. (f) List one set of vertices that form a pentagon. (g) List two sets of vertices that form scalene triangles (h) Which angles are congruent? List all congruent angle sets. 73. A contractor finds that the angle of elevation (the acute angle between the horizontal and the line of sight) to the top of a building is 46  35  as shown below. What is the measure of  QRP? 74. A forester from his vantage point on a hillside sees a puff of smoke in a valley below him. He measures the angle of depression (the acute angle between the horizontal and the line of sight) to the smoke from his position to be 21  13  15  (see figure). Find the measure of  PSF in degrees, minutes and seconds. F P S 21° 13  15  R Q P 46° 35  A B C 74° 69° 102° H G F E Chapter 2 17 A

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    75. A surveying party traversed the boundary of an area designated as a wetland. The interior angles formed were 90  12  , 132  46  , 102  16  , 120  40  and 94  . Determine the error, if any, in this traverse. (Remember: A traverse is the measurement of a polygonal region, made in a series of legs so that one leg is the start of the next leg and the beginning of the first leg connects with the end of the last leg.) 76. A certain mail order outlet uses a container in the shape shown below. Ignoring the flaps used to secure the sides together, describe and sketch the shape of the box when it is unfolded. 77. A right regular n-gon pyramid has 28 edges. (a) What n-gon is it? Show your work. (b) How many faces does the pyramid have? (c) How many vertices does the pyramid have? 78. A right regular n-gon prism has 64 vertices. (a) What n-gon is it? Show your work. (b) How many faces does the prism have? (c) How many edges does the prism have? 79. A can of beans is in the shape of a right circular cylinder. To recycle the can, the top and bottom are cut out and the remaining part of the can is cut and laid flat. Describe the shapes of the pieces of metal that remain for recycling. 80. Water is flowing along a stream at the rate of 1200 gallons per minute. What is this rate in liters per second? Round to the nearest hundredth. (Remember: There are four quarts in a gallon and 1.057 quarts in a liter.) 81. A foreign car has a gas tank that holds 45 liters of gasoline. Assuming the tank is empty and gas is selling for $1.23 a gallon, how much will it cost to fill this tank with gasoline? (Remember: There are four quarts in a gallon and 1.057 quarts in a liter.) 82. Light travels at 186,282 miles per second, one year is 365 days and Neptune is 2,697,000,000 miles from Earth. If a light year is the distance light travels in one year, how far is it from Neptune to Earth in light years? 83. If a car gets 32.6 miles to the gallon, what is its gas mileage in kilometers per liter? Round to the nearest hundredth. (Remember: There are four quarts in a gallon, 1.057 quarts in one liter, 2.54 centimeters in one inch and 5280 feet in one mile.) 18 Chapter 2

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    GEOMETRY Chapter Three True–False. Mark as true any statement that is always true. Mark as false any statement that is never true or that is not necessarily true. Be able to justify your answers. 1. If a right triangle has one leg that is 2 cm long and another leg that is 3 cm long, the area of the triangle is 3 cm 2 . 2. All rectangles with the same perimeter have the same area. 3. All parallelograms with the same area have the same perimeter. 4. The perimeter of a 3  by 1  rectangle is 8 sq.in. 5. The area of a right triangle with legs a and b is one-half the area of a rectangle with dimensions a and b. 6. The height of a trapezoid is the perpendicular distance between the bases. 7. The area of a circle with radius two is equal to its circumference. 8. If the sum of the squares of two sides of a triangle is equal to the square of its longest side, then it is a right triangle. 9. The area of a square with side s is less than the area of an equilateral triangle with side s. 10. For a triangle with side lengths a, b and c a 2  b 2  c 2 11. The area of a 45  –45  triangle with leg length 2 inches is the same as the area of a square with leg length inches. 12. Surface area may be expressed in centimeters. 13. The surface area of a rectangular prism is the sum of the areas of the six rectangles which form its faces. 14. The volume of any pyramid is the product of one-half the area of the base and its height. 15. The volume of a sphere is the product of  and four times the cube of the radius. 16. The volume of a right pyramid is of the volume of a right prism with the same height and the same base. 17. The volume of a sphere is of the volume of a right circular cylinder with the same diameter. Multiple Choice. Mark the letter of the single BEST response. Be sure to read all the choices for each problem before deciding. 18. The area of a rectangle 3  by 4  is the product of its perimeter and (a) 6/7 (b) n (c) 12/7 (d) 3/4 (e) None of these is correct. 1 3 1 3 1 2

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    19. Which of the following values most closely approximates  ? (a) 3 (b) 3 (c) 3.15 (d) 3.14 (e) 20. If the height, n, of a trapezoid equals the length of its shorter base and the longer base is twice the shorter base, then the area of the trapezoid is (a) 2n (b) 3n 2 (c) 1.5n 3 (d) 1.5n 2 (e) None of these is correct. 21. A regular octagon has a side of 12 cm and the perpendicular distance from the center to one of the sides is 14.5 cm. The area of the octagon is (a) 522 sq. cm (b) 696 sq. cm (c) 1044 sq. cm (d) 1392 sq. cm (e) 87 sq. cm 22. In a 45  –45  triangle, the hypotenuse is always this multiple of the leg lengths. (a) 2 (b) 2 (c) (d) 23. If ( a, b, c ) is a Pythagorean triple, then so is: (a) (3 a, 4 b, 5 c ). (b) (3 a, 3 b, 3 c ). (c) (6 a, 6 b, 6 c ). (d) Both (a) and (b) are correct. (e) Both (b) and (c) are correct. 24. If the legs of a right triangle are both n centimeters, then the length of the hypotenuse is (a) n 2 cm (b) 2n cm (c) n 2 cm (d) n cm (e) There is not enough information to determine the length. 25. In a 30  –60  right triangle the length of the longer leg is the product of the shorter leg and (a) (b) the hypotenuse. (c) (d) (e) 1 2 1 2 1 2 2 3 1 3 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 23 7 1 7 10 71 20 Chapter 3

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    26. A large rectangular flower planter is 4 feet  1.5 feet  9 feet. Potting soil comes in cubic yard bags. How many bags of potting soil are needed to completely fill the planter? (a) 108 bags. (b) 27 bags. (c) 4 bags. (d) 2 bags. 27. If the length of a lateral edge of a right square pyramid is 13 mm and a side of the base is 10 mm, then the slant height is (a) approximately 8.3 mm (b) 12 mm (c) mm (d) approximately 16.4 mm (e) Not enough information is given to find the slant height. 28. If the height of a right circular cone is three times the radius of the base, r, then the slant height of the cone is (a) r (b) 2r (c) 2r (d) r (e) There is not enough information given to find the slant height. 29. If the base of a right prism is a right triangle with legs 3 cm and 4 cm and the height of the prism is 10 cm, then the surface area of the prism is: (a) 60 cm 2 (b) 132 cm 2 (c) 120 cm 2 (d) 102 cm 2 30. If the base of a right prism is an equilateral triangle with side length 1 foot and the height of the prism is 1 foot, then the volume of the prism is: (a) cm 3 (b) cm 3 (c) cm 3 (d) cm 3 31. If the area of the base of a prism is 25 sq. in. and the height of the prism is 9 inches, then the volume of the prism is (a) 15 cu. in. (b) 75 cu. in. (c) 225 cu. in. (d) 112.5 cu. in. (e) None of these is correct 1 3 2 1 2 1 3 8 1 3 4 2 2 1 2 1 10 1 194 1 2 Chapter 3 21

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    32. If the slant height of a right square pyramid is equal to a side of its base, s, then the volume of the pyramid is (a) s 3 (b) s 3 (c) s 3 (d) s 3 (e) s 3 Fill in the Blanks. Complete each statement with a word or phrase that makes it true. 33. The ratio of the circumference of any circle to its diameter is . 34. The area of any region is always measured in units. 35. The area of a regular n -sided polygon with side length a and height h is . 36. If the legs of a right triangle are m and n, then the length of the hypotenuse is . 37. The formula SA   r(r  1) gives the surface area of a cone where r is the and 1 is the of the cone. 38. If r is the radius of a sphere, then the ratio of the surface area of the sphere to the volume of the sphere is . 39. The surface area of a right regular n -gon prism with base area A, height h and base edge length a is: 40. The volume of a sphere with radius r is: Writing. Write your answers concisely and completely. Feel free to use figures and/or tables to illustrate the points you are making. 41. Discuss the relationship between the area of a rectangle with dimensions a and b and the area of a right triangle with legs of a and b. 42. If a square with side length 1 unit and a circle with circumference 1 unit are drawn on top of each other, will the square be inside or outside the circle? Explain. 43. Describe how the height of an equilateral triangle with side length s is found 44. Describe how the hypotenuse of a 45  –45  triangle compares to the length, s, of the legs of the triangle. Exercises/Problems. 45. The radius of a circle is 6 cm. The side of a square is 6 cm. Compare their areas. Then compare the perimeters of the circle and the square. 1 3 1 6 2 2 1 2 2 3 1 6 22 Chapter 3

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    46. By counting squares and parts of squares, inside or outside the figures, find the area of the following figures: 47. Determine the perimeter of each figure: 48. Determine the area of each figure: 49. A triangle has sides 17 cm, 12 cm and 11 cm. If the altitude to the 11 cm side is 11.95 cm, what is the length of the altitude to the 17 cm side? Round your answer to one decimal place. 50. The area of a circle is 36  sq. ft. What is its circumference to the nearest hundredth of a foot? a. . b 2.5 cm 7 cm 4 cm 2.5 cm 4 cm c. . d 1.64 cm 1.5 cm 10 cm 8 cm 20 cm a. . b 8 cm 2 cm 5.5 feet 3 feet c. . d 2.4 meters 2.1 cm 2.25 cm 2.05 cm 2.1 cm 1.95 cm 1.6 cm a. b. 1 square unit Chapter 3 23

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    51. Find the area of the shaded figure. Assume the curves are semicircles. Round your answer to the nearest hundredth. 52. The length of one leg of a right triangle is 7  more than the length of the shorter leg. If the hypotenuse is 7  more than the longer leg, find the area of the triangle. 53. If the length of the hypotenuse of a 45  –45  right triangle is 7.3 cm, find its area and perimeter to the nearest hundredth. 54. If the hypotenuse of a 30  –60  right triangle is 7 , cm find the exact value of the lengths of the legs. 55. What is the surface area of the rectangular prism shown below? 56. Find the surface area of the right square pyramid shown below. 57. Find the volume of a cone with slant height of 10  and base radius of 6  . Applications. 58. A wheat farmer plants a field with winter wheat. The shape of the field is shown below. If 43,560 square feet equals one acre, how many acres are in this field? Round your answer to the nearest tenth. 430  25  100  70  80  840  1843  1370  1648  24 mm 20 mm 13.5  9.1  9.1  1 2 1 3 14.2 cm 6.8 cm 24 Chapter 3

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    59. In the following sequence of regular pentagons, each toothpick represents one unit. Determine the perimeter if we arrange n such regular pentagons in a row. 60. The distance from our Sun to Earth is approximately 93,000,000 miles. This distance is also called an Astronomical Unit (AU). The distance from the Sun to Mars is 0.4 AU. Assuming the orbit of Mars to be roughly circular, what is the distance in miles traveled by Mars in one revolution around the Sun in miles? 61. A tile in the shape of a regular hexagon is shown. How many tiles will be required to cover a rectangular counter top area 3.4  by 3  ? (Remember: One inch equals 2.54 centimeters.) 62. A sprinkler is set in a corner of a rectangular lawn 10 feet by 25 feet. If the maximum distance the sprinkler can reach is ten feet, what percent of the lawn will be watered from this position? 63. Find the perimeter and the area of the following figure. Assume the lightly shaded inner triangle is an equilateral triangle with each side length 1 unit and assume the curves are semicircles that cover half of each side of the triangle. Give the exact answers. 64. Find the area of the following figure. Assume the triangle is a right isosceles triangle and the curve is a semicircle. Show your work. 12 feet 12 feet 135 ° 2 feet 2 feet 10 feet 9.01 cm 10.4 cm Chapter 3 25

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    65. The following equilateral triangle with side length 4 cm has three circular regions surrounding it. What is the area of the shaded portion of the figure? The radius of each circle is 1 cm. Show your work. 66. The following figure is composed of one large and three small semicircles. Find the shaded area enclosed by the four semicircles. Give the exact answer. 67. For the following two concentric circles, what would the radius of the larger circle need to be in order for the area of the inner circle to equal the shaded area between the two circles? 68. A dart board consists of concentric circles and line segments as shown. The radius of the bull’s eye is 4 inches and each ring surrounding the bull’s eye is 3 inches wide. (a) What is the area of the bull’s eye? (b) What percent of the total area is worth ten points? 6 6 5 5 2 8 8 2 10 10 10 9 9 R r  12 feet 12 feet 1 cm 4 cm 1 cm 1 cm 26 Chapter 3

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    69. A baseball diamond is a square 90  on a side. If a ball is hit to the shortstop who is halfway between second base and third base, how far must he throw the ball to put the runner out at first base? 70. Two trucks leave a city at the same time. One heads south at an average rate of 58 mph. The second truck heads due east at an average rate of 63 mph. After 1 hour and 35 minutes, how far apart are the trucks? 71. A 12-foot ladder will be used to paint the exterior of a house. The foot of the ladder needs to be 3.7 feet from the base of the house. What is the maximum height the ladder will reach? 72. Find a formula for the distance d between opposite sides of a regular hexagon in terms of the length of one side x . 73. A ramp is designed as shown below. The surfaces are to be made of 1  plywood. Then the top surface is to be covered with a non-slip coating. How much plywood is required? What area will be covered with the non-slip coating? 74. Assuming Pluto and Earth are spheres and their respective diameters are 2500 km and 12,756 km, how many times greater is the surface area of Earth than the surface area of Pluto? 75. A cardboard box needs to be 3 feet long, 1.5 feet wide, and 2 feet deep. Allowing an additional 5% for seams and waste, how much cardboard will be required to build this box? 76. The following right regular hexagonal pyramid has an edge length of 6 cm on the hexagonal base and the height of the pyramid is 8 cm. (a) What is the surface area of the pyramid? (b) What is the volume of the pyramid? 8cm 6 cm 4  1.5  12  2  d x 1st 2nd 3rd Home Chapter 3 27

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    77. A right circular cylinder with a wall that is 1 cm thick is dipped into paint. How many square centimeters of paint are needed to cover the cylinder (inside and out)? The radius of the inner wall, r, of the cylinder is 4 cm, the radius of the outer wall, R, of the cylinder is 5 cm and the height of the cylinder is 10 cm. Give the exact answer. 78. Jupiter is the largest planet in the solar system with a diameter of 142,796 km. Pluto is the smallest planet in the solar system with a diameter of 2500 km. Assuming these planets to be roughly spherical, how many times greater is the volume of Jupiter than the volume of Pluto? 79. A board foot is the volume of a piece of wood 1 foot by 1 foot by 1 inch. A deck is designed as shown below. One inch by six inch boards are to be used for the surface of the deck. How many board feet are required for this part of the job? 80. A cylindrical cooling sleeve for a beverage is filled with liquid for freezing. The sleeve is 1 cm thick; the inner radius of the sleeve is 4 cm, the outer radius of the sleeve is 5 cm and the height of the sleeve is 10 cm. How much liquid is needed? Give the exact answer. R Outer R  5 cm Inner r  4 cm h  10 cm r 3  3  8  20  r R Outer R  5 cm Inner r  4 cm h  10 cm 28 Chapter 3

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    GEOMETRY Chapter Four True–False. Mark as true any statement that is always true. Mark as false any statement that is never true or that is not necessarily true. Be able to justify your answers. 1. If a statement is true then its contrapositive must also be true. 2. If p q and q r then r p. 3. The statement “if p then q” can be read as “q only if p.” 4. If a statement is false, then its converse must also be false. 5. If  MAD   ROC , then  DMA   ORC . 6. If two triangles are congruent, then there are exactly three pairs congruent parts—the corresponding angle pairs. 7. In  ABC , if  then  BAC   ACB . 8. If  ,  and  ABC   XYZ then  ABC   XYZ 9. In  ABC,  ABC   CBA by corresponding parts (C.P.) 10. If bisects , then  . 11. To construct a copy of , we begin by drawing a new line. 12. The construction of a perpendicular bisector of a segment requires three arcs and one line. 13. The construction of the bisector of an angle requires three arcs and one line. Multiple Choice. Mark the letter of the single BEST response. Be sure to read all the choices for each problem before deciding. 14. If an argument reads If  ABC is an equilateral triangle with perimeter 3 s units then each edge of  ABC is s units long. If each edge of an equilateral triangle is s units long then its height is units. If the height of an equilateral triangle is units, then its area is units 2 . Then we can deduce (a) If  ABC is an equilateral triangle with perimeter 3 s units then its area is units 2 (b) If the area of  ABC is units 2 then it is an equilateral triangle (c) If each edge of an equilateral triangle is s units long then its area is units 2 . (d) All of these are valid conclusions. (e) Both (a) and (c) are correct. 1 3 s 2 4 1 3 s 2 4 1 3 s 2 4 1 3 s 2 4 1 3 s 2 1 3 s 2 AB BP AP AB PQ YZ BC XY AB BC AB Q Q Q

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    15. If an argument reads p q q r r s, then we can conclude (a) s (b) p s (c) p (d) s p (e) No conclusion can be reached. 16. The statement: “If ABCD is a square, then ABCD is a parallelogram,” can be rewritten as (a) If ABCD is a parallelogram, then it is q square. (b) ABCD is a square if and only if it is a parallelogram. (c) If ABCD is not a square, then it is not a parallelogram. (d) All squares are parallelograms. (e) All parallelograms are squares. 17. If an argument reads: If , then  PQR is a right angle. If  PQR is a right angle, then  PQR is a right triangle. If  PQR is a right triangle with  Q the right angle, then ( PQ ) 2  ( QR ) 2  ( PR ) 2 . then we can conclude (a) . (b) If then ( PQ ) 2  ( QR ) 2  ( PR ) 2 . (c) If  PQR is a right angle then ( PQ ) 2  ( QR ) 2  ( PR ) 2 . (d) All of these are valid conclusions. (e) Both answers (b) and (c) are correct. 18. The symmetric property of congruence allows us to conclude that if  , then (a)  . (b)  . (c)  . (d) All of these follow from the symmetric property. (e) Both (b) and (c) follow from the symmetric property. 19. If  ABC   XYZ and  XYZ   RST, then (a)  ABC   RST . (b)  XYZ   RST . (c)  . (d) All of these are valid conclusions. (e) Both (b) and (c) are correct. 20. In two right triangles, if two pairs of sides and one pair of angles are congruent then (a) The triangles are congruent by the Pythagorean Theorem and HL. (b) The triangles are congruent by HL. (c) The triangles are congruent by SSS. (d) Both (a) and (c) are correct. (e) There is not enough information to tell if the triangles are congruent. XY AB AB AB PQ PQ AB PQ PQ AB QR  PQ QR  PQ QR  PQ Q Q Q Q Q 30 Chapter 4

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