7 th Grade Study guide IV Partial Remember to practice the constructions that are not part of this guide.

Transcription

1 7 th Grade Study guide IV Partial Remember to practice the constructions that are not part of this guide. 1. Which figure shows one point? a. S R c. D C b. Q d. F G 2. Which name describes the line? G C F K D a. b. c. d. 3. The floors in an apartment building belong to different. a. rays b. segments c. planes d. points 4. Use the diagram. and. D C A B E F H G a. will intersect c. are parallel b. have point C in common d. will never meet In the diagram a b. Use the diagram to answer the question. (Diagram not to scale.)

2 a b If and, what is the measure of? a. 20 b. 60 c. 40 d In the figure,. Find the measure of. B D E 50 o G A C a. 40 b. 110 c. 50 d Classify the triangle by its sides and angles in in in.

3 not drawn to scale a. equilateral, acute c. equilateral, right b. scalene, right d. isosceles, acute Name all the quadrilaterals that have the given property. 8. two pairs of parallel sides and all angles congruent a. trapezoid c. rhombus, square b. rectangle, square d. rhombus 9. The school band was scheduled to march in the annual parade, but the band s starting point was changed the day before the parade. The band director calls three band members. Each band member calls three other band members. Then these band members each call three members. How many band members, including the band director, are notified of the new starting point? a. 12 members b. 13 members c. 20 members d. 40 members In the figures,. A Z C D X Y 10. Name the congruent angles. a. c. b. d. 11. If CD = 26 mm, what is XY? a. 28 mm b. 25 mm c. 4 mm d. 26 mm 12. In the floor tile pattern, the figure is a square and the two triangles are congruent. What is the length of? 5 inches T I A L a. 5 in. b. 20 in. c. 10 in. d. 7.1 in. Write a congruence statement for the pair of triangles. 13. B R 5 ft 5.5 ft 5 ft 5.5 ft Z Y O 3 ft 3 ft W

4 a. by SAS c. by SSS b. by SSS d. by SAS Students on a field trip at an amusement park were asked their grade in school. The table shows the results of the survey. Grade Number of Students 4th 45 5th 25 6th 60 7th 50 8th 60 9th What percent of the students surveyed were in 9th grade? Round your answer to the nearest percent, if necessary. a. 45% b. 9% c. 8% d. 16% 15. Suppose you want to show the data in a circle graph. Find the measure of the central angle for 5th grade. Round your answer to the nearest degree, if necessary. a. 9 b. 32 c. 16 d People in the mall were surveyed about their favorite type of movie. The table shows the results of the survey. Make a circle graph of the data. Type of Movie Percent Comedy 40% Drama 25% Action 20% Science fiction 15% a. c. Comedy Drama Sci-Fi Comedy Action Sci-Fi Action Drama b. d. Comedy Drama Comedy Drama Action Sci-Fi Action Sci-Fi

5 Graph the image of for the translation units down a. c. b. d. 18. Use arrow notation to describe the translation of point P( 5, 4) to point P ( 8, 7). a. c.

6 b. d. 19. Write a rule to describe the translation of to. a. c. b. d. 20. Jim and Kim are designing a mural on a wall of the school. They placed a coordinate plane on the wall. Quadrilateral is a translation of quadrilateral ABCD. Use the table of translations to find the coordinates of point B. Quadrilateral ABCD Quadrilateral A( 5, 3) ( 8, 1) B(?,?) ( 11, 5) C( 4, 1) ( 7, 3) D( 7, 3) ( 10, 1) a. ( 9, 8) b. ( 14, 3) c. ( 13, 2) d. ( 8, 7) 21. Write a description of the rule. a. translation 2 units to the right and 7 units up b. translation 2 units to the right and 7 units down c. translation 2 units to the left and 7 units up d. translation 2 units to the left and 7 units down 22. Graph with vertices R(1, 7), S( 6, 3), and T(4, 4) and its image after a reflection over the x-axis.

7 a. c. b. d. 23. Graph with vertices R(6, 6), S(3, 6), and T(0, 3) and its image after a reflection over the y-axis. a. c.

8 b. d. 24. Graph and its image after a rotation of 90 counterclockwise about the origin. a. c.

9 b. d. 25. Find the coordinates of the image of a triangle with vertices A(0, 5), B( 9, 0), and C( 6, 2) under a rotation of 90 clockwise about the origin. a. (0, 5), (9, 0), (6, 2) c. (0, 5), (9, 0), ( 6, 2) b. (5, 0), (9, 0), ( 2, 6) d. ( 5, 0), (0, 9), (2, 6) 26. What are the coordinates of the point ( 10, 3) after a 180 clockwise rotation about the origin? a. ( 3, 10) b. ( 10, 3) c. (10, 3) d. (10, 3) 27. Use the diagram. C F D a. Name three points. b. Name four different segments. c. Write two other names for. d. Name three different rays. 28. Use the diagram. G P U T S Q V W R

10 a. Name four segments that intersect. b. Name three segments that are parallel to. c. Name four segments that are skew to. 29. Draw two parallel segments. Then draw a line that intersects the parallel segments. 30. A shipping box measures 12 inches by 16 inches by 20 inches. a. Make a sketch of the box. Label each vertex of the prism. b. Name two parallel segments in your drawing. c. Name two skew segments in your drawing. 31. Construct a segment congruent to. C D 32. Construct a segment 2 times the length of. R S 33. Construct the perpendicular bisector of the segment. X Y 34. Construct the angle bisector of the angle. S 35. Elsie is making a quilt using quilt blocks like the one in diagram.

11 a. Copy the figure and draw all lines of symmetry. How many lines are there? b. Does the quilt square have rotational symmetry? If so, what is the angle of rotation? 36. Gabriel is making a quilt using this pattern for each quilt square Drawing not to scale. a. If = 68, what is? Explain your method for finding the angle measure. b. If, what do you know about? Explain your reasoning. 37. A fish pond at the local park is a regular hexagon. a. Write a formula for the perimeter of the pond in terms of the length of a side. Explain your formula. b. Each side has a length of 7.5 feet. Find the perimeter of the pond. c. Suppose the designer of the pond wants to make another regular hexagonal pond with a perimeter of 57.6 feet. Find the length of one side of the pond. Explain your method. 38. Solve the problem by drawing a diagram. Jolene is making a design using a circle. She places a number of points on the circle, and plans to connect each point with each other point with pieces of string. a. Complete the table to show the number of pieces of string for different numbers of points around the circle. Number of Points Number of Strings

12 9 10 b. Write a formula for finding the number of strings S needed for any number of points n. Explain your reasoning. 39. Bill s Grill is a concession stand at the beach.the table shows how many of each type of food item were sold during one week in June. Food Number Sold Hamburgers 600 Hot dogs 100 Chicken sandwiches 300 a. Find the central angle represented by each food item in a circle graph. Round answers to the nearest degree. Explain your method. b. Draw a circle graph for the data. c. Suppose another food item, a burrito, is added to the menu, and 600 were sold that week. How does the percent for hamburgers change? Show how you found the answer. 40. Emily is designing a logo for a company using a coordinate plane. First, she draws Triangle 1 with vertices A( 1, 3), B( 4, 5), and C( 3, 0). Second, she draws Triangle 2 which is the reflection of Triangle 1 over the x-axis. Third, she draws Triangle 3 which is the reflection of Triangle 2 over the y-axis. Finally, she draws Triangle 4 which is the reflection of Triangle 3 over the x-axis. a. Draw Emily s design on a coordinate plane. b. Describe the relationship between Triangle 1 and Triangle 4 in terms of a reflection. c. Suppose one vertex of Triangle 1 is (a, b). What will be the corresponding vertex in Triangle 4? Explain. 41. Three triangular gardens are being planned for a new park. Sergio has placed his plans on a coordinate plane to help him determine the locations for the gardens. The diagram shows the location of the first garden named. a. First, Sergio reflects over the line y = x to form the second garden. Copy the diagram and draw. Describe how the coordinates for relate to the coordinates for. b. Next, Sergio reflects over the x-axis to form the third garden. Draw on your drawing for part a. Describe how the coordinates for relate to the coordinates for.

13 c. Can you name a rotation that would move to. If so, what is the angle of rotation? 42. The diagram shows several streets in Maryville. Apple Street and Pine Street are parallel and 1st Avenue intersects them. The measure of = 77. Find. Explain your method for finding the measure. 1st Avenue 2 Apple Street 1 Pine Street Diagram not to scale. 43. Justyne built the huge kite shown in the diagram. The two pieces of wood represented by and meet forming four right angles. A B 3 ft 3 ft F 3 ft D 6 ft C a. Classify by its sides and angles. Explain. b. Which triangle is identical in size and shape to? c. Classify by its sides and angles. Explain. 44. Carson made this computer graphic. He knows that RO = PO = AO = GO.

14 a. Explain why the four triangles are congruent. b. What is the measure of? Explain your reasoning. 45. Darius is marking out some lines on the gym floor for a game. He has marked the segment shown. Describe how he can construct a 135 angle with vertex at A and one side as. A is the midpoint of. M A G 46. Eric is making a graphics design on a coordinate plane. He graphed a pentagon with vertices A(1, 0), B(4, 0), C(4, 2), D(2.5, 4), and E(1, 2). Then he translated the pentagon 5 units left and 3 units up. a. Write the coordinates for the translated pentagon. Describe how you found the coordinates. b. Graph ABCDE and on the same coordinate plane. c. Describe a translation that would move ABCDE to pentagon located entirely in Quadrant IV. 47. Rachel is using the figure in designing a video game. y x a. Copy the figure. Then reflect the figure over the line y = x. b. Use your drawing from part a. Reflect the figure from part a over the y-axis. c. Compare the original figure to the figure in part b. Describe it as a rotation. d. If one point of the original figure has coordinates (a, b), what will be the coordinates of the corresponding point on the figure in part b? Explain your reasoning. 48. Find the area of the rectangle. 14 ft 8 ft a. 44 b. 64 c. 196 d. 112

15 Find the area of the parallelogram cm 12 cm 6 cm a. 60 cm 2 b. 216 cm 2 c. 108 cm 2 d. 72 cm 2 Find the area of the triangle in. 9.5 in. 3 in. a. 27 in. 2 b in. 2 c in. 2 d in Find the area of a circle with radius 8.1 m to the nearest square unit. a. 66 m 2 b. 824 m 2 c. 206 m 2 d. 51 m Find the area of the figure to the nearest square unit. 3 cm 8 cm a. 74 cm 2 b. 49 cm 2 c. 37 cm 2 d. 125 cm 2 Find the surface area of the space figure represented by the net.

16 53. 5 cm 5 cm 7 cm 8 cm 4 cm a. 124 cm 2 b. 110 cm 2 c. 150 cm 2 d. 164 cm Find the surface area of a rectangular prism that is 16 inches long, 12 inches wide, and 5 inches high. a. 960 in. 2 b. 689 in. 2 c. 714 in. 2 d. 664 in A conical tent made of canvas has a base that is 24 feet across and a slant height of 14 feet. To the nearest whole unit, what is the area of the canvas, including the floor? Use 3.14 for π. a. 2,864 ft 2 b. 716 ft 2 c. 980 ft 2 d. 1,507 ft George made a conical hat to match his costume for a party. The hat has a slant height of 14 inches and a base circumference of 16 inches. The cone is open at one end. To the nearest square unit, what is the lateral area of the hat? a. 112 in. 2 b. 162 in. 2 c. 56 in. 2 d. 2,813 in Find the surface area of the sphere to the nearest square unit. Use a calculator. 2 in. a. 13 in. 2 b. 50 in. 2 c. 6 in. 2 d. 3 in You are designing a new container for powdered laundry detergent. You are considering a cylindrical container with a diameter of 14 inches and a height of 18 inches. Find the volume of this container to the nearest cubic unit. Use a calculator. a. 1,100 in. 3 b. 882 in. 3 c. 11,084 in. 3 d. 2,771 in Find the volume of a can of soup that has a height of 16 cm and a radius of 5 cm. Use 3.14 for π.

17 a. 1,256.0 cm 3 b cm 3 c. 4,019.2 cm 3 d cm You cut square corners with side lengths that are whole numbers from a piece of cardboard with dimensions 20 inches by 30 inches. You then fold the cardboard to create a box with no lid. Which of the following dimensions will give you the greatest volume? a. 12 in. by 22 in. by 4 in. c. 14 in. by 24 in. by 2 in. b. 10 in. by 20 in. by 5 in. d. 10 in. by 24 in. by 6 in. Find the volume of the cone to the nearest cubic unit. Use a calculator. 61. height 8 cm; radius 15 cm a. 1,885 cm 3 b. 1,461 cm 3 c. 5,655 cm 3 d. 22,619 cm Find the missing dimension. Round to the nearest unit. Use 3.14 for π. h 6 cm V = Height =? a. 2.7 cm b cm c. 4 cm d. 1.3 cm 63. h 10 in. V = 200 in. Height =? a. 6.7 in. b. 12 in. c. 10 in. d. 6 in. Find the surface area of the prism.

18 64. a. 6,720 m 2 b. 1,662 m 2 c. 1,872 m 2 d. 3,360 m The Donaldsons bought the unusually shaped lot shown below on which to build a new house. 68 ft 78 ft 77 ft a. What is the area of the lot? b. If the Donaldsons build a rectangular house with length 40 feet and width 30 feet, what percent of the area of the lot will be covered by the house? 66. The vertices of a parallelogram are A( 1, 2), B(3, 2), C(4, 3), and D(0, 3). a. Draw the parallelogram. b. Find the area of the parallelogram. 67. For the figure, describe the base(s), if any, and name the figure. 68.

19 An architect is designing a half-spherical dome above a circular fountain and path. The architect wants the dome to be only over the fountain and path. The total diameter of the fountain with surrounding path is 45 feet. a. What is the height of the cover? Explain. b. What is the surface area of the cover? Show your work. 71. The Flying Eagle Wild Bird and Game Preserve is shaped approximately as in the diagram. 3 mi 7 mi 4 mi a. Find the area of the preserve in square miles. Explain how you find the area. b. There are 640 acres in one square mile. What is the area of the preserve in acres? Explain how you find the number of acres. c. Additional land is available for the preserve. The preserve could be expanded to another trapezoid-shaped region with the same height but with bases of length 7 miles and 5 miles. If the preserve is expanded to this new area, what is the percent of increase in area? Explain how you find the percent. 72. At the Magic Garden, a rose garden is being designed as shown. The outer figure is a square with side length of 116 feet. a. What is the diameter of one circle? Explain how you find the diameter. b. The roses are to be planted in the four circles. The rest of the space will be covered by wood chips. What is the area of the surface that will be covered by wood chips? Explain how you find this area. 73. For a particular square pyramid, the length of a side of the base is 8 cm and the slant height is 12 cm. For another square pyramid, the length of a side of the base is 12 cm and the slant height is 8 cm. a. Find the surface area of each pyramid. Explain your steps for finding the surface area. b. Explain why the surface areas are different while both have dimensions of 8 cm and 12 cm. 74. Megan has a large packing box shaped as a cube with a volume of 216 cubic feet. a. What is the side length for the cubical box? Explain how you find the length.

20 b. Megan would like to design a box that is a rectangular prism, but not a cube. What are a possible length, width, and height that she could use if she wants this box to have the same volume as the cube? Explain how you find the dimensions. 75. Stetson has a ball that is a sphere with a radius of 9 cm. He plans to pack it tightly in a box that will be 2 cm wider than the sphere. a. Describe the box using its dimensions. b. Find the volume of the spherical ball and the box. Explain how you find both volumes. c. What percent of the volume of the box will be occupied by the ball? Explain how you find the percent. 76. A new circular fountain being designed for a park has a diameter of 30 feet. a. Find the surface area of the water in the fountain. Explain how you find the area. b. Suppose the designer of the fountain decides that the surface area of the water in the fountain should be 450 square feet. Find the diameter of this fountain. Explain how you find the diameter. 77. A bin for storing harvested grain on a farm is shaped like a cylinder. Describe the bases and the lateral surface of the bin. 78. Jessica is designing a cylindrical storage container for lawn chemicals. She first designed a cylinder with radius 12 inches and height 16 inches. a. What is the surface area of this container? Explain how you find the surface area. b. Jessica is considering changing the container by doubling either the radius or the height of the container. Will doubling the radius of the original container or doubling the height of the original container cause the greater percent of increase in surface area from the original? Explain your method for answering this question. 79. Jodi is making some decorations for a graduation party. A diagram of a decoration is shown. She plans to hang 30 of these figures on wires across the room. What will be the total surface area of the 30 figures? Explain how you find the surface area. 80. Bridger City has a cylindrical tank for storing water used by the residents. The tank has a diameter of 30 feet and a height of 25 feet. a. What is the volume of the tank? Explain how you find the volume. b. One cubic foot of water is about 7.5 gallons of water. About how many gallons of water are in the tank? Explain how you find the number of gallons. c. The city would like to build an additional tank with a volume of 150,000 gallons. Find a possible diameter and height for the new tank. Explain the method you use to find the dimensions. 81. Spaceship Earth at Epcot Center in Florida is a 180-foot geosphere. a. Estimate its volume by assuming it is a sphere with diameter 180 feet. Explain how you estimate the volume. b. Estimate its surface area by assuming it is a sphere. Explain how you estimate the surface area.

21 c. Explain why the volume has a greater numerical value than the surface area even though the volume formula contains the value and the surface area formula contains the value 4. Simplify the square root. 82. a. 16 b. 0.4 c. 40 d Which of these sets of numbers contains no irrational numbers? a.,, 8.15 c. b. d., 84. Which of these sets of numbers contains no rational numbers? a. c. 6,, 4 b., 13 d. 85. The surface area of the top surface of the water in a circular swimming pool is about 206 square feet. Estimate the radius of the pool, to the nearest foot. a. about 14 feet b. about 8 feet c. about 11 feet d. about 10 feet 86. Which number is rational? a. b. c. d. 87. Which number is irrational? a. b. c. d. 88. In the given right triangle, find the missing length. 24 m c 10 m Not drawn to scale a. 28 m b. 26 m c. 25 m d. 27 m 89. Two flag poles in front of the Court House are 12 ft tall. The distance from the top of one pole to the base of the other as shown in the diagram is 20 ft. What is the distance between the two flag poles?

22 a. 16 ft b. 23 ft c. 18 ft d. 15 ft The lengths of two sides of a right triangle are given. Find the length of the third side. Round to the nearest tenth if necessary. 90. leg: 20 m; hypotenuse: 25 m a m b. 8.9 m c. 32 m d. 15 m 91. Which of the following could NOT be the lengths of the sides of a right triangle? a. 9 ft, 12 ft, 15 ft c. 4 cm, 7.5 cm, 8.5 cm b. 5 in., 10 in., 15 in. d. 1.5 m, 2 m, 2.5 m Find the distance between the two points. Round to the nearest tenth if necessary. 92. ( 2, 1) and (3, 5) a. 4.1 b. 2.2 c. 3.6 d Find the perimeter of. Round to the nearest tenth. a b c d Find the midpoint of the segment with the given endpoints. 94. D(1, 2) and E( 3, 6) a. ( 7, 10) b. ( 2, 2) c. ( 1, 4) d. (4, 64)

23 95. A(3, 5) and C(2, 9) a. ( 1, 5.5) b. (0.5, 7) c. (2.5, 2) d. (8, 7) 96. Find the midpoint of. a. (3, 1) b. (1, 1) c. (10, 4) d. (4, 4) 97. Which function is a quadratic function? a. c. b. d. 98. The graph of which function passes through the point ( 2, 2)? a. b. c. d. Graph the function. 99. a. c.

24 b. d. Identify the number as rational or irrational. Explain Is a triangle with sides of length 3 m, 4 m, and 5 m a right triangle? Explain Is a triangle with sides of length 6 ft, 21 ft, 23 ft a right triangle? Explain A new park is placed on a coordinate plane to help in locating important features of the park. The diagram below shows the vertices of the quadrilateral defined by the fences of the park. a. At the midpoint of each side, the Park Commission will have an entrance gate. Find the midpoints of the sides. Call the midpoint of M, the midpoint of O, the midpoint of X, and the midpoint of Y. b. Find the perimeter of the quadrilaterals PARK and MOXY. If necessary, round to the nearest tenth. c. Compare the perimeters of PARK and MOXY For the function y = 2x + 1, make a table with integer values of x from to 2. Then graph the function.

25 106. Skip bought a lot on which he wants to build a house. The lot is a square with an area of 10,000 square feet. a. Let s be the length of the side of the lot. Write an equation that models this situation. Explain the equation. b. Find the length of one side of the lot. c. Skip plans to build a house with a length of 39 feet and a width of 43 feet. What percent of the lot will be covered by the house? Round to the nearest whole percent. Explain your method for solving this problem Vance built a triangular sand box for his daughter. The measures of the lengths of the sides are 21 feet, 28 feet, and 37 feet. a. Explain how you know the sand box is not in the shape of a right triangle. b. What should the lengths of the sides be in order for the sand box to form one right angle? Explain your reasoning Orlando is surveying at a mine. He made the diagram below to help him find the distance x across a mining pit. Z 23 ft A x B 138 ft 115 ft O Not drawn to scale P a. Assume that. Write a proportion that includes the length x. Explain the proportion. b. Find the length x. Explain your method for finding this length. c. Find the length of. Explain your method. d. Find the length of. Explain your method. e. What is the length of? 109. The City Commission wants to construct a new street that connects Main Street and North Boulevard, as shown in the diagram below.

26 New Street 9 mi Main St. 4 mi North Blvd. Not drawn to scale a. What will be the length of the new street? Explain how you found this length. b. The construction cost has been estimated at $140 per linear foot. Estimate the cost for constructing the street. Explain your method for finding the estimate On Echo s Farm, individual fields are laid out in rectangles like WBSA on the coordinate plane below. The units marked are in feet. a. What is the area of the field WBSA? Explain your method for finding the area. b. In order to avoid stepping on crops, you walk from the shed (S) to the well (W) by first walking to point A. Using this route, how far is it from the shed to the well? Explain your method for finding the distance. c. What is the shortest distance between the shed and the well? Explain your method for finding this distance.

27 7th Grade Study Guide Answer Section 1. B 2. A 3. C 4. D 5. B 6. C 7. A 8. B 9. D 10. C 11. D 12. A 13. C 14. D 15. B 16. A 17. A 18. C 19. B 20. D 21. D 22. C 23. A 24. D 25. D 26. C 27. a. Answers may vary. Answers should include three of these points: C, D, F, G. b. c. 28. a. d. Answers should include three of these rays:. b. c. 29. Answers may vary. Sample: C D A B

28 30. a. Answers may vary. Sample: H J F G 16 inches C D A 20 inches B 12 inches b. Answers may vary. Sample: and c. Answers may vary. Sample: and 31. N O ) A B C ) ) 32. X R S 33. Y R Z 34. S T

29 35. a. 36. The quilt square has 4 lines of symmetry. b. The quilt square has rotational symmetry of 90?. [4] a. Since and are adjacent angles that lie on a straight line, the angles are supplementary, or have measures with a sum of 180. You can use this fact to find. + = 180 and are supplementary = 180 Replace with = Solve for. = 112 The measure of is 112. b. If, then =. Explanations may vary. Sample: + = 180 and are supplementary. + x = 180 Let = x. + = 180 and are supplementary. x + = 180 Since and = x, then = x. + x = x + Since both = 180, they are equal. + x x = x + x Subtract x from each side. = Simplify. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanation [1] a correct answer with no explanation 37. [4] a. Let P represent the perimeter of the pond and s represent the length of one side. Since a regular hexagon has six sides of equal length, a formula is P = 6s. b. The perimeter is 45 feet. c. To find the length of one side of the hexagon, substitute 57.6 for P in the formula and solve for s. P = 6s formula for perimeter 57.6 = 6s Substitute 57.6 for P. Divide each side by = s Simplify. The length of one side of the pond is 9.6 feet. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanation

30 [1] a correct answer with no explanation 38. [4] a. Number of Number of Points Strings b. Let the number of points be n. From each vertex you can connect n points with n 1 points, but those are counted twice so you have to divide by two. The formula is or. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanation [1] a correct answer with no explanation 39. [4] a. To find the central angle represented by each food item, first find the total number of food items sold. Then use proportions to find the measures of the central angles. Total items sold = = 1,000 hamburgers: ; r = 216 hot dogs: ; r = 36 chicken sandwiches: ; r = 108 b. Foods Sold at Bill s Grill Hamburgers Hot Dogs Ch. Sand. c. To find the percent for hamburgers using the new item, you must find a new total for the items. Then use a proportion to find the percent. Total items sold = = 1,600 hamburgers: ; r 37.5 The percent of food items represented by hamburgers decreased from 60% to 37.5% of the food items sold. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanation [1] a correct answer with no explanation

31 40. [4] a. b. If you reflect Triangle 1 over the y-axis, you will have Triangle 4. c. Triangle 1 and Triangle 4 have the same y-coordinates, but the x-coordinates are opposites, so the corresponding vertex for Triangle 4 is ( a, b). [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanation [1] a correct answer with no explanation 41. [4] a. If a vertex of (b, a). b. has coordinates (a, b), then the corresponding vertex for the reflection will be

32 If a vertex of has coordinates (a, b), then the corresponding vertex for the reflection will be (a, b). c. If you rotate 90 clockwise about the origin, you will get. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanation [1] a correct answer with no explanation ; Explanations may vary. Sample: Let the angle supplementary to and on the opposite side of 1st Avenue be. Since and are supplementary and their measures add to 180, then = , or 103. and are corresponding angles so they are congruent and their measures are equal. So, = a. has two congruent sides with measures of 3 feet, so it is an isosceles triangle. It has a right angle at F so it is a right triangle. b. is identical in size and shape to. c. has no congruent sides, so it is a scalene triangle. It has a right angle at F so it is a right triangle. 44. a. By looking at the vertices of the quadrilateral and their coordinates, you can see that the quadrilateral is a square, so GR = RA = AP = PG. For and, they are congruent by SSS because RA = PA, RO = PO, and the triangles share. The same reasoning can be used for each pair of triangles. b. If, then the corresponding angles are congruent. Since and are supplementary and equal, each has to measure 180 2, or Open the compass to more than half the length of. Put the compass tip at M. Draw an arc intersecting. With the same compass setting, repeat from point G. Label the points of intersection of the two arcs as X and Y. Draw. should also contain A as this segment bisects. is a right angle since is the perpendicular bisector of. Now, bisect. Put the compass tip at A. Draw an arc that intersects the sides of. Label the points of intersection H and J. Put the compass tip at H. Draw an arc. With the same compass setting, repeat with the compass tip at J. Make sure the arcs intersect. Label the intersection of the arcs T. Draw. Now, measures 90 + (90 ), or a. To find the new coordinates, subtract 5 from each x-coordinate and add 3 to each y-coordinate. ( 4, 3), ( 1, 3), ( 1, 5), ( 2.5, 7), and ( 4, 5) b.

33 c. Answers may vary. Sample: Translate ABCDE 6 units down. 47. a. y x y = x b. y x c. If you rotate the original figure 90 counterclockwise about the origin, it would match the figure in part b.

34 d. If a point on the original figure has coordinates (a, b), then the corresponding point on the figure in part b will be ( b, a). To determine this, you can see that one point on the original figure is ( 2, 1). The image of that point for part b is ( 1, 2). The x-coordinate of the figure in part b is the opposite of the y-coordinate of the original figure.the y-coordinate of the figure in part b is the x-coordinate of the original figure. 48. D 49. C 50. B 51. C 52. B 53. D 54. D 55. C 56. A 57. A 58. D 59. A 60. B 61. A 62. C 63. D 64. C 65. a. 5,236 ft 2 b. about 23% 66. a. b. 20 square units 67. The base is a rectangle. The figure is a pyramid. 68. The base is a circle. The figure is a cone. 69. There are no bases. The figure is a sphere. 70. a. The height is 22.5 feet because the diameter of the sphere is 45 feet and the dome is one-half of a sphere. b. S.A. = The surface area is the area of a sphere. =

35 71. = 3, The surface area of the dome is about 3,179 square feet. [4] a. To find the area of the preserve, use the formula for the area of a trapezoid. Use the formula for the area of a trapezoid. Replace h with 7, b 1 with 4, and b 2 with 3. = Simplify. = 24.5 The area of the preserve is 24.5 square miles. b. To find the area in acres, multiply the number of square miles by = 15,680 There are 15,680 acres in the preserve. c. First find the area of the enlarged preserve using the same steps as in part a. Use the formula for the area of a trapezoid. Replace h with 7, b 1 with 7, and b 2 with 5. = Simplify. = 42 The enlarged preserve would have an area of 42 square miles. Next, find the percent of increase in area by writing a fraction with the amount of increase as the numerator and the original area as the denominator. Then write the fraction as a percent. 72. = = 71.4% The percent of increase in area is 71.4%. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanation [1] correct answers with no explanation [4] a. The length of one side of the square is 116 feet. There are two circles across the square, so the diameter of one circle is or 58 feet. b. To find the area of the wood chips, find the area of the square and subtract the area of four circles. Use the formula for the area of a square. = Substitute 116 for s. = 13,456 Simplify. Use the formula for the area of a circle. Multiply by Substitute 3.14 for π and 29 for the radius. = 10,563 Simplify. area of square area of 4 circles = 13,456 10,563 or 2,893

36 73. The area of surface covered by wood chips is about 2,893 square feet. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanation [1] correct answers with no explanation [4] a. For each pyramid, use the formula for surface area of a pyramid. first pyramid S.A. = L.A. + B formula for surface area of a pyramid = L.A. = = = 256 second pyramid S.A. = L.A. + B = L.A. = = p = 4(8) and l = 12 formula for surface area of a pyramid p = 4(12) and l = 8 = 336 The surface area of the first pyramid is 256 cm 2 and the surface area of the second pyramid is 336 cm 2. b. Answers may vary. Sample: For the second pyramid, the side length of 12 is squared, which causes that surface area to be greater than the other pyramid where 8 is squared. [3] one mathematical error or correct answers with error in reasoning [2] two mathematical errors or correct answers with several errors in reasoning or explanations [1] correct answers with no explanation 74. [4] a. To find the side length of the cube, you must find a number that equals 216 when cubed. By Try, Test, Revise, you find that the side length is 6 feet. b. Answers may vary. Sample answer: Since 216 is divisible by 6, the height could be 6. Then Bh = B(6) and B = 36. Two numbers that multiply to 36 are 9 and 4. So a possible box could have a height of 6 feet, length of 4 feet, and width of 9 feet for a volume of 216 cubic feet. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanation [1] correct answers with no explanation 75. [4] a. The box is a cube where the side length is the diameter of the ball plus 2 cm. So, the side length is 2(9 cm) + 2 cm, or 20 cm. b. To find the volumes, use the volume formulas for a sphere and a rectangular prism. sphere Use the volume formula. Replace r with 9. 3,052 Simplify. cube V = Bh Use the volume formula. = (20 20)(20) Replace all dimensions with 20 since the box is a cube.

37 = 8,000 Simplify. The volume of the ball is about 3,052 cm 3 and the volume of the box is 8,000 cm 3. c. To find the percent, write a fraction with the volume of the ball as the numerator and the volume of the box as the denominator. Then write the fraction as a percent or about 38% The ball occupies about 38% of the volume of the box. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanation [1] correct answers with no explanation 76. a. To find the surface area of the water in the fountain, use the formula for the area of a circle. The radius of the circle will be 30 2 = 15. = = 225π The surface area of the water is about square feet. b. Answers may vary. Sample reasoning: To find the radius of the new fountain, use the formula for the area of a circle. Solve the equation for r. Then multiply the radius by 2 to get the diameter. Substitute 450 for A. Divide each side by π. Simplify. Simplify. Now you need to find a value for r such that r r 143.3, which is about 12. The diameter is approximately 12 feet 2, or 23.9 feet. 77. The bin has two parallel circular bases. The lateral surface is curved, but if cut open perpendicular to a base and laid flat, it would be a rectangle. 78. a. To find the surface area, use the formula for surface area for a cylinder. S.A. = L.A. + 2B formula for surface area of a cylinder S.A. = 2πrh + 2(πr 2 ) Substitute the formula for the L.A. and 2 circles. 2(3.14)(12)(16) + 2(3.14)(12) 2 r = 12, h = 16 = 1, = 2, The surface area of the original container is about 2110 square inches. b. First, find the surface area of each new container. Container with radius doubled S.A. = L.A. + 2B formula for surface area of a cylinder S.A. = 2πrh + 2(πr 2 ) Substitute the formula for the L.A. and 2 circles. 2(3.14)(24)(16) + 2(3.14)(24) 2 r = 24 (doubled), h = 16 = 2, , Container with height doubled S.A. = L.A. + 2B formula for surface area of a cylinder S.A. = 2πrh + 2(πr 2 ) 2(3.14)(12)(32) + 2(3.14)(12) 2 r = 12, h = 32 (doubled) = 2, Substitute the formula for the L.A. and 2 circles.

38 3316 Now, find the percent of increase in surface area for each new container. Container with radius doubled 1.86 or 186% Container with height doubled = = 0.57 or 57% Doubling the radius causes the greater percent of increase in surface area. 79. First, find the surface area of one decoration. The figure is two cones placed together, so you need to find the lateral area of two cones and no bases. S.A. = 2 L.A. = 2 (πrl) formula for lateral area of a cone 2 (3.14)(3)(7) r = 3 and l = 7 = The surface area of one figure is about cm 2. Now, multiply the surface area by 30 for 30 figures = 3,956.4 The surface area is about 3,956 cm a. To find the volume of the tank, use the formula for volume of a cylinder. You need to find the radius of the tank which is 30 feet 2, or 15 feet. V = Bh Use the formula for volume. V = πr 2 h B = πr 2 h 3.14(15 ) 2(25) Replace r with 15 and h with 25. = 17,662.5 The volume is about 17,662.5 cubic feet. b. To find the approximate number of gallons, multiply the number of cubic feet by , = 132, The volume is about 132,469 gallons. c. Answers may vary. Sample: If the new tank is to have a volume of 150,000 gallons, then that would be 150,000 gallons 7.5 gallons per cubic foot, or 20,000 cubic feet. Using the formula for volume of a cylinder, then 20,000 = πr 2 h. Choose a height, such as 25 feet. Then 20,000 = πr 2 (25) or, dividing both sides by 25π, 255 r 2. Now you need a value for the radius such that r r 255, so r is about 16. If the radius is 16, then the diameter is 32. So a new water tower could have a diameter of 32 feet, a height of 25 feet, and a volume of 20,000 cubic feet or 150,000 gallons. 81. a. To estimate the volume, use the formula for the volume of a sphere. Use a radius of or 90. Use the volume formula. Replace r with 90. = 3,052,080 Simplify. An estimate for the volume is 3,000,000 cubic feet.

39 b. To find the surface area, use the formula for the surface area of a sphere. Use a radius of or 90. Use the formula for surface area. Replace r with 90. = 101,736 Simplify. The surface area is about 100,000 square feet. c. For the volume, you cube the radius while you only square it for the surface area. That causes the numerical value of the volume to be greater. 82. D 83. B 84. A 85. B 86. D 87. C 88. B 89. A 90. D 91. B 92. D 93. B 94. C 95. C 96. B 97. C 98. A 99. B 100. Rational; it is a terminating decimal Irrational; it is the square root of a number that is not a perfect square yes; 103. no; 104. a. M(0, 3), O(4, 1), X(0, 1), Y( 4, 1) b. perimeter of PARK: 24 units; perimeter of MOXY: 17.9 units c. The perimeter of PARK is about 1.3 times the perimeter of MOXY x 2x + 1 = y (x, y) 2( 2) + 1 = 9 ( 2, 9) 2( 1) + 1 = 3 ( 1, 3) 0 2(0) + 1 = 1 (0, 1) 1 2(1) + 1 = 3 (1, 3) 2 2(2) + 1 = 9 (2, 9)

40 106. [4] a. An equation is = 10,000. In the equation, is the area of the square lot since each side measures s units. The area of the square is 10,000. b. The length of one side is feet, or 100 feet. c. The area of the house is square feet, or 1,677 square feet. To find the percent of the lot covered by the house, write and solve a proportion. = Write a proportion. n = 167,700 Write cross products. = Divide each side by 10, n 17 Simplify. The house covers about 17% of the lot. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanations [1] correct answers with no explanation [4] a. You can use the Converse of the Pythagorean Theorem to show that the sides do not form a right triangle. The longest side should be used for the length of the hypotenuse. = Use the Pythagorean Theorem. Replace a with 21, b with 28, and c with Simplify The sides of the sand box do not form a right triangle. b. Answers may vary. Sample: In order for the sides of the sand box to form a right triangle, the side lengths must satisfy the Pythagorean Theorem. Choose two side lengths and find a third that will work. Pick 21 feet and 28 feet for the lengths of the legs. = Use the Pythagorean Theorem. = Replace a with 21 and b with 28. = Simplify = Add.

41 108. = c Find the positive square root of each side. 35 = c The sand box could have sides of length 21 feet, 28 feet, and 35 feet. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanations [1] correct answers with no explanation [4] a. If, then corresponds to and corresponds to. You can write a proportion using these corresponding parts of the form. b. = Write a proportion x = Write cross products. 138x = 2,645 Use the Distributive Property. 23x = 2,645 Subtract 115x from each side. x = 115 Divide each side by 23. The distance x across the pit is 115 feet. c. To find the length of, you can use the Pythagorean Theorem since you know the lengths and, which are both 115 feet. = Use the Pythagorean Theorem. = Replace a and b with ,225 13,225 = Simplify. 26,450 = Add. = c Find the positive square root of each side c Round to the nearest tenth. The length of is about feet. d. To find the length of, you can use the Pythagorean Theorem since you know the lengths of two legs. is 138 feet long and is, or 138 feet long. = Use the Pythagorean Theorem. = Replace a and b with ,044 19,044 = Simplify. 38,088 = Add. = c Find the positive square root of each side c Round to the nearest tenth. The length of is about feet. e. The length of is about, or 32.6 feet. [3] one mathematical error or correct answers with incomplete explanations [2] two mathematical errors or correct answers with errors in explanations [1] correct answers with no explanation a. To find the length of the new street, use the Pythagorean Theorem. The missing length to be found is the hypotenuse of the right triangle. = Use the Pythagorean Theorem. = Replace a with 4 and b with 9. = Simplify. = 97 Add.

42 110. c = Find the positive square root of each side. c 9.8 Round to the nearest tenth. The length of the street is about 9.8 miles. b. To estimate the cost of the street, find the number of feet in 9.8 miles and multiply that number by the cost per foot. 9.8 miles = 9.8 miles 5,280 feet per mile = 51,744 feet The cost is about 51,744 feet $140 per foot, or $7,244,160. a. First, you need to find out the units for each tick mark on the graph. Since there are 6 marks from 0 to 150, each unit is 25 feet. You can see that from A to W is 3 units and from A to S is 2 units, so the dimensions of the field are 75 feet by 50 feet. That area then is square feet, or 3,750 square feet. b. To find the distance, you can see that there are 3 units from W to A and 2 units from A to S. That is a total of 5 units. If each unit is 25 feet, then the total distance is feet = 125 feet. c. The shortest distance between the two points can be found by using the Distance Formula. W has coordinates ( 25, 25) and S has coordinates (25, 50). d = Use the Distance Formula. d = Replace with ( 25, 25) and with (25, 50). d = Simplify. d = Find the exact distance. d 90.1 Round to the nearest tenth. The distance is about 90.1 feet.