d = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 Student Name: Date: Teacher Name: Sunil Dudeja Score:

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1 Geometry EOC (GSE) Quiz Answer Key Equations and Measurement - (MGSE9 12.G.GPE.4) Use Coordinates For Theorems, (MGSE9 12.G.GPE.5 ) Prove Slope Criteria, (MGSE9 12.G.GPE.6) Find The Point, (MGSE9 12.G.GPE.7 ) Use Coordinates For Perimeter/area, (MGSE9-12.G.GMD.4) Identify Shapes, (MGSE9-12.G.MG.1) Describe Objects, (MGSE9-12.G.MG.2) Density Concepts, (MGSE9-12.G.MG.3) Geometric Methods 1) Student Name: Teacher Name: Sunil Dudeja Date: Score: Each unit on the grid stands for one mile. Determine two ways to calculate the distance from Josie's house to Annie's house. A) Distance Formula and Slope Formula B) Midpoint Formula and Slope Formula C) Distance Formula and Midpoint Formula D) Distance Formula and Pythagorean Theorem You can use the Distance Formula and Pythagorean Theorem. Both formulas will give the same distance from Josie's house to Annie's house. d = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 a 2 + b 2 = c 2 1/17

2 2) On a coordinate grid, the movie theater is located at (0,0) and the mall is located at (4,3). If the bowling alley is located at the midpoint between the theater and the mall, what is the approximate distance from the bowling alley to the mall? (Note: 1 unit equals 1 mile) A) 1.3 miles B) 1.5 miles C) 2 miles D) 2.5 miles Solution: 2.5 miles. The first step required for solving this problem is to calculate the midpoint between the theater and the mall (2,1.5). Then use the distance formula to calculate the distance from the bowling alley to the mall. d = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 M = ( x 1 + x 2 2, y 1 + y 2 ) 2 2/17

3 3) Two points are shown on the graph. What is the distance between the two points? A) 6 units B) 8 units C) 10 units D) 12 units 10 units The points are (-5, -3), (1, 5). D = ( ) D = ( ) D = (100) D = /17

4 4) y = 2x + 3 y = 2x - 5 What is the BEST description for the lines represented by the equations? A) skew B) parallel C) vertical. D) intersecting The solution is parallel. The slope in both equations is 2. Therefore, the lines are parallel. 5) Line A: y = 1 2 x + 2 Line B: y = 1 2 x + 7 Line C: y = 2x + 4 Line D: y = 1 2 x Which lines are perpendicular? A) A and B B) A and C C) B and C D) A and D Lines B and C are perpendicular. The equations shown are in the form y = mx + b where m represents the slope of the line. Lines B and C have slopes that are opposite reciprocals of each other and are therefore perpendicular. 4/17

5 6) What is the best description for the lines? A) parallel B) vertical C) perpendicular D) the same line Equation 1: x - 3y = 9 Equation 2: y = -3x + 3 Perpendicular lines have slopes that are opposite reciprocals of each other. That is, they have different signs and are reciprocals. When equation 1 is put in slope-intercept form, it becomes y = 1 3 x - 3. Its slope is 1. Equation 2 is already in slope-intercept form. 3 Its slope is and -3 are opposite reciprocals. The two lines are perpendicular. 7) Find the point, M, that is five-sixths of the distance from A(-7, 2) to B(-1, -4). A) (-1, -3) B) (-2, -3) C) (-1, -4) D) (-2, -4) (-2, -3) The coordinates of M are (x m, y m ), where x m = (-1 - (-7)) and y m = (-4-2) /17

6 8) Find the point, M, that divides segment AB into a ratio of 5:5 if A is at (0, 15) and B is at (20, 0). A) (35, 10) B) (20, 10) C) (10, 7.5) D) (17.5, 5) (10, 7.5) The sum of the ratio numbers (5+5) is 10, so M is 5 10 = 1 2 of the distance from A to B. The coordinates of M are (x m, y m ), where x m = = (20-0) and y m = (0-15). 2 9) Find the point, M, that divides segment AB into a ratio of 3:1 if A is at (-4, -2) and B is at (4, -10). A) (8, 2) B) (4, -2) C) (-2, 4) D) (2, -8) (2, 4) The sum of the ratio numbers (3+1) is 4, so M is [[3/4] of the distance from A to B. The coordinates of M are (x m, y m ), where x m = = (4 - -(4)) and y m = (-10 - (-2)). 4 10) What is your location on the coordinate plane if you walked one-third of the way from home to school in a straight line? A) (3, 11 2 ) B) (3, 11 3 ) C) ( 9 2, 11 3 ) D) ( 9 2, 11 2 ) (3, 11 3 ) 6/17

7 (mx 2 + nx 1 ) (m + n), (my 2 + ny 1 ) (m + n) Where the point divides the segment internally in the ratio m:n ((1)(7) + (2)(1)), (1 + 2) ((1)(1) + (2)(5)) (1 + 2) = 9 3, 11 3 = (3, 11 3 ) 7/17

8 11) On the coordinate grid of a map, Josie's house is located at (2,7). Her school is located at (-5,5). If each map unit equals one mile, what is the approximate distance from her house to school? A) 2.83 miles B) 4.79 miles C) 7.28 miles D) miles Solution: 7.28 miles. The distance formula is used to solve this problem. d = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 12) What is the area of the rectangle shown in the graph? A) -28 units 2 B) -3 units 2 C) 30 units 2 D) 54 units 2 Find the lenght and the width by counting the distance. The negatives do not matter in distance. So the area = 9 6 = 54 units 2 8/17

9 13) The drawing represents A) the intersection of a plane and a cone. B) the intersection of a plane and a prism. C) the intersection of a plane and a sphere. D) the intersection of a plane and a cylinder. The drawing represents the intersection of a plane and a sphere. The plane in the diagram is slicing horizontally through the sphere. 14) A plane intersects a rectangular prism as shown. Describe the cross-section. A) circle B) rectangle C) trapezoid D) triangle The solution is triangle. As you can see in the diagram, the plane slices through the rectangular prism at an angle and forms a triangular cross-section. 9/17

10 15) If this triangle is translated backwards through space, what three-dimensional figure will be formed? A) Square pyramid. B) Triangular prism. C) Rectangular prism. D) Triangular pyramid. When the triangle is translated backwards through space, a triangular prism is formed. The original triangle serves as one base and when the triangle stops translating, the second triangle is the other base of the prism. The three edges of the triangle extend to form the faces of the triangular prism. 16) The batteries in the diagram best resembles what geometric solid? A) cone B) cylinder C) pyramid D) sphere The solution is cylinder. A cylinder has two circular bases as seen in the diagram of the batteries. 10/17

11 17) The vertices on the snowflake pictured form what kind of geometric figure? A) decagon B) hexagon C) pentagon D) triangles The vertices form a hexagon. If you draw a line around the perimeter of the snowflake, you will notice it has 6 sides, a hexagon. 18) Which triangle is a regular polygon? A) All triangles B) An equiangular triangle C) triangle D) triangle An equiangular triangle The sides of a regular polygon are all equal. The sides of an equiangular triangle are all equal. Therefore, it is a regular polygon. 19) At Macon County High School, a classroom is 400 ft 2. After the teacher sets up her desk, computer, and podium, there are 250 ft 2 left for student desks. If each desk takes up 8 ft 2 of room, how many desks can fit in the classroom? A) about 19 B) about 25 C) about 31 D) about 50 Divide 250 by 8. There is enough room for about 31 desks. 11/17

12 20) Given an 8 foot length of tree trunk with a radius of 3 feet, what is the volume of the tree trunk section? (to the nearest whole ft 3 ) A) 75 ft 3 B) 151 ft 3 C) 226 ft 3 D) 452 ft ft 3 Use the formula for the volume of a cylinder to model the volume of the tree trunk. V = πr 2 h V = π(3 2 )(8) V = 72π ft ft 3 21) Alex determines that there are 5 chocolate candies per cubic inch. If there are 150 chocolate candies in a jar, what is the volume of the jar? A) 30 in 3 B) 75 in 3 C) 500 in 3 D) 750 in 3 Divide 150 by 5. The volume of the jar is 30 in /17

13 22) Given that the population density of Georgia is 170 people per square mile and 10,100,000 people live in Georgia, what is the area of the state? (to the nearest hundred mi 2 ) A) 50,000 mi 2 B) 50,500 mi 2 C) 58,800 mi 2 D) 59,400 mi 2 59,400 mi 2 Population Density = Number of People Land Area 170 = 10,100,000 Area Area = 59,412 23) A jewelry designer is working with a design that is a regular hexagon inscribed in a circle. He needs to cut a diamond to fit into each of the triangles shown. What is the measure of the indicated angle? A) 30 B) 45 C) 60 D) 360 Since this is a regular hexagon you know that each angle has the same measure. There are 360 in a circle and divide that by the number of angles, 6. The measure of the indicated angle is /17

14 24) A plane is to be loaded with bottles of water and medical supplies to be sent to victims of an earthquake. Each bottle of water serves 10 people and each medical kit aids six people. The goal is to maximize z, the total number of people helped, where z = 10x + 6y, and x is the number of bottles and y is the number of medical kits. Using the constraints of the situation (plane weight and volume capacities) the shaded region is in the graph is obtained. Which vertex of the region represents the solution to the maximization problem? A) (0, 6000) B) (4000, 0) C) (2000, 4000) D) (6000, 8000) (2000, 4000) is correct. Test each of the 4 vertices [(0, 0), (0, 6000), (4000, 0), and (2000, 4000)] in the z function to get the largest answer. 14/17

15 25) An open box will be made from a rectangular piece of cardboard that is 8 in. by 10 in. The box will be cut on the dashed red lines, removing the corners, and then folded up on the dotted lines. The box needs to have the MAXIMUM volume possible. How long should the cuts be? A) 1.5 in. B) 5.8 in. C) 52 in. D) 80 in. The height, width, and length of the box will be (x)(8-2x)(10-2x). The volume of the box will be 4x 3-36x x = 0 Use a graphing calculator to graph the polynomial. Use the maximum feature to see the greatest volume the box can have. Be sure to set your window accordingly and think about what constraints are on the box. The maximum volume for the box is 52 in 3. The cut would be 1.5 in. long. This is where the graph crosses the x-axis. 26) A steel plant has two sources of ore, source A and source B. In order to keep the plant running, at least three tons of ore must be processed each day. Ore from source A costs $20 per ton to process, and ore from source B costs $10 per ton to process. Costs must be kept to no more than $80 per day. Moreover, Federal Regulations require that the amount of ore from source B cannot exceed twice the amount of ore from source A. If ore from source A yields 300 lbs of steel per ton, and ore from source B yields 400 lbs of steel per ton, how many tons of ore from each source should be processed each day to maximize the amount of steel produced? A) 1 ton from source A, 2 tons from source B B) 2 tons from source A, 4 tons from source B C) 3 tons from source A, 0 tons from source B D) 4 tons from source A, 0 tons from source B 2 tons from source A, 4 tons from source B is correct. The three inequalities are a+b 3, b 2a, and 20a + 10b 80. They form a quadrilateral (with the x-axis) having vertices (1,2), (2,4), (3,0), (4,0). Test each of these in the production function, P = 300a + 400b to get the answer. 15/17

16 27) An open box will be made from a rectangular piece of cardboard that is 8 in. by 10 in. The box will be cut on the dashed red lines, removing the corners, and then folded up on the dotted lines. What is the MAXIMUM possible volume for the box? A) 1.5 in 3 B) 5.8 in 3 C) 52 in 3 D) 64 in 3 The height, width, and length of the box will be (x)(8-2x)(10-2x). The volume of the box will be 4x 3-36x x = 0 Use a graphing calculator to graph the polynomial. Use the maximum feature to see the greatest volume the box can have. Be sure to set your window accordingly and think about what constraints are on the box. The maximum volume for the box is 52 in 3. The cut would be 1.5 inches long. This is where the graph crosses the x-axis. 16/17

17 28) Farmer John needs to build a new watering trough for his horses. He has laid out a diagram on grid paper so he knows how much material to buy. If each unit mark represents 1 foot, what is the total square footage of material he needs to buy? A) 2 ft 2 B) 8 ft 2 C) 12 ft 2 D) 16 ft 2 The total surface area is 8 ft 2 2 squares 1 ft 2 = 2 ft 2 3 rectangles 2 ft 2 = 6 ft 2 2 ft ft 2 = 8 ft /17