Georgia Online Formative Assessment Resource (GOFAR) 8th Grade Geometry Review

Transcription

1 8th Grade Geometry Review Name: Date: Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 28 )

2 1. Line segments PS and QT intersect at point R. Point R is a vertex of right triangle RPQ. What is the measure of angle PQR? Show your work and explain your answer. What is the length of segment RQ? Show your work. 2. A city planner uses a grid to show points P and Q. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (2 of 28 )

3 What is the equation of the line including points P and Q? Show your work and explain your answer. The city planner wants to know the distance between points P and Q. Find the distance, in units, between points P and Q. Show your work and explain your answer. 3. The figure includes squares JKLM, LNPQ, and MQRS, and right triangle LQM. Square JKLM has an area of 169 square inches. Side MQ has a length of 12 inches and side LQ has a length of 5 square inches. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (3 of 28 )

4 Find the areas of LNPQ and MQRS, and describe a relationship among the areas of the three squares. How does the relationship support the Pythagorean theorem? Construct a figure similar to the one above that includes three squares making one right triangle. Label the side length of each square. Find the area of each square. Show your work and explain your answer. 4. Triangles DEF and JKL are shown. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (4 of 28 )

5 What information is provided to show triangle DEF is similar to triangle JKL? What series of transformations could be used to obtain triangle JKL from DEF? 5. Right triangle JKL is graphed on a coordinate grid. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (5 of 28 )

6 If the triangle is reflected across the y-axis, what would be the vertices of the reflected triangle? Label the new vertices as and What are the lengths of line segments JK and KL? Part C What is the length of line segment Part D What is the relationship of triangles JKL and Show your work. Explain your answer. 6. Rectangle A is shown on the coordinate grid. Rectangle A is dilated by a factor of 0.5 about the center point The new rectangle formed is rectangle B. Create a coordinate grid similar to the one shown above. Graph and label rectangle B on the coordinate Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (6 of 28 )

7 grid you created. Rectangle B is reflected across the y -axis to form rectangle C. Write the coordinates of each vertex of rectangle C. Explain how the coordinates change when rectangle B is reflected across the y -axis to form rectangle C. Part C Rectangle A is rotated counter-clockwise about the origin to form rectangle D. Graph and label rectangle D on the same coordinate grid you created in. Part D Is rectangle D similar to rectangle B? Explain your answer. 7. Mr. Novak draws a figure on a coordinate grid. He begins to construct a new figure congruent to the figure shown by points and as shown in the coordinate grid. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (7 of 28 )

8 Mr. Novak used two different transformations to create the new figure. Based on the location of points and what would be the coordinates of and Describe a sequence of transformations which would result in the new figure being congruent to the original figure based on point and Part C Create a similar figure, but not congruent, to Mr. Novak s new figure in quadrant I. What are the coordinates of and Part D Describe the sequence of transformations you used to create the figure in Part C. 8. The figure includes squares JKLM and PQRS. The lengths of each square side are labeled. What expression, using a and b, represents the area of square JKLM? Explain your answer. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (8 of 28 )

9 Write two expressions representing the area of square PQRS. One expression should include only c. One expression should include only a and b. Part C Write an equation that relates the two expressions from. Part D Simplify the equation for c. What does the equation prove? Explain your answer. 9. Liam graphed triangle DEF on the coordinate grid. What are the coordinates of triangle DEF after it is translated 1 unit to the right and 6 units down? What are the coordinates of the translated triangle from after it is rotated 180 about the origin? Part C Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (9 of 28 )

10 Liam says the distance in units between the original point D and the corresponding vertex after the translation and reflection is greater than the distance in units between the original point D and the corresponding vertex after only the translation. Is Liam correct? Explain your answer. 10. A cone with a diameter of 16 centimeters and a slant height of 17 centimeters is shown. What is the height, h, of the cone in centimeters? Show your work. What is the volume of the cone in terms of? Show your work. Part C Jacob designs a party hat with the same diameter as the cone, but with a height 5 centimeters taller. How does the volume of the cone compare to the volume of the party hat Jacob designs? Show your work and explain your answer. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (10 of 28 )

11 11. Use the right triangle shown to answer each part. Write an expression, in simplest form, to represent the length of the missing side of the triangle. Show your work. The missing side of the triangle has a value of 768. Using the expression from, determine the value of x. Show your work. Part C Using the value of x from, calculate the side lengths of the triangle. Show your work. Part D Using the side lengths of the triangle, explain how the use of squares with the same side lengths as the right triangle can be used to prove the triangle is a right triangle. 12. Fill in the blank. All are either similar or congruent to each other. A)circles B)triangles C)rectangles D)parallelograms 13. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (11 of 28 )

12 The triangles shown are congruent. Which of the following two sides MUST be proportional? A)CM and SV B)CM and MT C)CT and VU D)CT and MT 14. Look at the triangles below. Which of these triangles are most likely similar? A)1 and 2 B)1 and 3 C)2 and 3 D)3 and Susan has two boxes. Each box is 12 cm high, 12 cm long, and 12 cm wide. Which statement describes Susan's boxes? A)The boxes are congruent, but not similar. B)The boxes are similar, but not congruent. C)The boxes are similar and congruent. D)The boxes are only similar. 16. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (12 of 28 )

13 Alyssa is building a birdhouse. She needs a right triangle for the roof. Which triangle should Alyssa use? A) B) C) D) 17. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (13 of 28 )

14 A right triangle is shaded in the diagram. The area of two squares is shown. What is the area of the third square? A)21 ft2 B)35 ft2 C)441 ft2 D)1,241 ft2 18. What is the measure of the missing side of the sail? A)18 feet B)20 feet C)22 feet D)24 feet 19. What is the length of a diagonal of a rectangle with length 12 and width 5? A)7 B)13 Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (14 of 28 )

15 C)17 D) Using the measures shown in the sketch, what is the length of the section of the ladder from the point where it rests on the ground to the point where it touches the house? A)4.8 ft. B)7 ft. C)17 ft. D)23 ft. 21. To secure a telephone pole, a steel cable will be attached to a ring on the pole 12 feet above the ground. If the cable is 15 feet long, how far from the base of the pole will the other end of the cable be anchored? A)3 feet B)9 feet C)12 feet D)15 feet 22. Sue left her house traveling due west towards the store. After 50 yards she traveled due north 120 yards to Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (15 of 28 )

16 the store. When she left the store she cut across the field and traveled along a straight path. How much shorter was the path Sue took home then the path she took to the store? A)40 yards B)61 yards C)70 yards D)109 yards 23. Seth can either ride his bike or walk to school. He can ride at 5 mph for 0.6 miles on Elm Street and then 0.8 miles on Maple Street, or he can walk at 2 mph on the walking trail. How much longer should it take Seth to walk to school than to ride his bike? A)0.2 hours B)0.22 hours C)0.28 hours D)0.4 hours 24. The locations of a student's house and two markets are shown on this coordinate plane. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (16 of 28 )

17 Each unit represents one mile. To the nearest 0.1 mile, how much closer is Farmer's Garden than Prod A)0.4 mile B)0.7 mile C)1.8 miles D)2.9 miles 25. This coordinate plane shows the locations of two campsites. The campground owner plans to install a water tap at a location that is equidistant from the two camp A) Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (17 of 28 )

18 B) C) D) 26. Quadrilateral is shown on this coordinate plane. What is the length, to the nearest unit, of diagonal A)8 B)10 C)12 D) On this coordinate plane, line represents a bike path. Each unit represents 2 yards. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (18 of 28 )

19 A deer is grazing at point A)8 yards What is the shortest distance between the deer and the bike path? B)10 yards C)12 yards D)16 yards 28. Ryan's house lies 6 miles due east of Adrian's house. Adrian's house is 8 miles due south of Frank's house. What is the shortest distance from Frank's house to Ryan's house? A)8 miles B)10 miles C)14 miles D)24 miles 29. The shape of the moon is most like a A)cone. B)cylinder. C)pyramid. D)sphere. 30. A tennis ball is most like a A)circle. B)cone. C)cylinder. D)sphere. 31. Which of the following items is best described as a sphere? A)pizza B)soda can C)cereal box D)tennis ball 32. What is the volume of the cylinder? Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (19 of 28 )

20 A) cm3 B) cm3 C) cm3 D)2, cm3 33. The height of this tennis ball can is 26 cm, and its radius is about 4.2 cm. What is the BEST estimate for the volume of this can? A)400 cm3 B)900 cm3 C)1,500 cm3 D)1,800 cm3 34. If you wanted to find out how much water you could pour into a cylindrical bucket, which type of measurement would you want to calculate? A)perimeter of the bucket B)area of the bucket C)volume of the bucket D)circumference of the bucket 35. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (20 of 28 )

21 Four friends bought ice cream cones. Gabriella's cone had a radius of 2.5 inches, and a height of 3.5 inches. Julie's cone had a radius of 2.5 inches, and a height of 4 inches. Robert's cone had a radius of 2.2 inches, and a height of 5 inches. Luke's cone had a radius of 2.2 inches, and a height of 6 inches. Whose cone could hold the most ice cream? A)Gabriella B)Julie C)Robert D)Luke 36. Leonardo is making gift packages. He has a cone shaped package and a cylindrical package. The cone and cylinder have the same radius and height. He wants to use the package with the greatest volume. Which package should Leonardo use? A)the cone B)the cylinder C)The cone and cylinder have equal volumes. D)There is not enough information to tell. 37. A beverage company wants to see if a new cylinder can design will hold more juice than the old rectangu lar prism design. They know that the radius of the new can is 2 inches and the height is 5 inches. The length of the old carton is 4 inches, the width is 3.2 inches and the height is 7.7 inches. Which container holds more juice? A)The containers hold equal amounts of juice. B)The volume of the new container is 62.8 cubic inches C)The new container holds more juice than the old container. D)The old container holds more juice than the new container. 38. A cylindrical container has a radius of 6cm. The container is holding 255 cm3 of water. Rounded to the nearest quarter inch, what is the MINIMUM height of the container? A) cm B)7 cm C) cm D) cm 39. Mary needs to buy a cylindrical trashcan liner that will fit into her trashcan. She knows that her trashcan has a volume of 300 cubic units and the diameter of her trashcan is 16 units. How high does the liner need to be to fit into her trashcan. Round your answer to the nearest whole number. A)0 units high Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (21 of 28 )

22 B)1 unit high C)2 units high D)5 units high 40. Ashley is making a skateboard ramp in the shape of a triangular prism. The figure shows three dimensions of the ramp. What is the missing dimension in inches, x, of Ashley s skateboard ramp? A)22 B)26 C)27 D) The figure shows part of a stepladder with two steps. Each step is parallel to the ground and attached to a diagonal rod. Which conclusion is true based on the given information and angle relationships? A)The value of x is 80 because vertical angles are congruent. B)The value of x is 80 because adjacent angles are congruent. C)The value of y is 100 because vertical angles are supplementary. D)The value of y is 100 because alternate exterior angles are supplementary. 42. The coordinate plane shows the coordinates of the vertices of Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (22 of 28 )

23 is translated 2 units to the left and 5 units up to create of What are the coordinates of the vertices Is congruent to Explain your answer. is dilated using a scale factor of 2 to create coordinates of the vertices of The center of dilation is the origin. What are the Is congruent to Explain why or why not. Part C is reflected over the x-axis to create What are the coordinates of the vertices of Is congruent to Explain why or why not. Part D Compare answer. from Parts A, B, and C. Which triangles, if any, are similar? Explain your Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (23 of 28 )

24 43. The diagram shows the side measures and two angle measures of The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles of the triangle. Based on this information, what is the measure of the exterior angle at vertex F? Show your work. What is the measure of the interior angle at vertex F? Explain how you got your answer. Part C Triangle DEF will be rotated clockwise about vertex D and then dilated by a scale factor of 3. This transformation will take D to E to and F to Sketch and label its vertices, angle measures, and side lengths, including the appropriate units of measure. Show your work or explain your work. Part D What is the relationship between and Explain your answer. 44. Karen constructed a vase using a right cone made of glass and a right cylinder made of metal. When the glass cone rests inside the metal cylinder, the vertex of the cone lies on the center of the circular base. The diagram shows the vase and the dimensions of its parts. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (24 of 28 )

25 What is the volume of the glass cone? Show your work. Use the dimensions shown in the diagram to determine how many times greater the volume of the cone is than the volume of the cylinder. Show your work or explain your answer. Part C When the vase is constructed, what fraction of the cone s volume is not inside the cylinder? Show your work. 45. The diagram shows a right rectangular prism with a base diagonal and a diagonal between opposite vertices of the lower and upper bases. Micah and Kiara need to find the exact length of the longest diagonal of the prism, d cm, for a homework assignment. Micah and Kiara both made at least one error during their calculations. Micah s solution method is shown. Explain each step of Micah s solution method and correct any errors he made. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (25 of 28 )

26 Kiara s solution method is shown. Explain each step of Kiara s solution method and correct any errors she made. Part C What is the exact value of d? Show or explain your work. 46. A large picture window is made from 7 pieces of glass: 2 large squares, 2 large isosceles triangles, and 3 small isosceles triangles. Adjacent figures share sides and vertices, as shown in the diagram of the window. The area of square BCGF is 36 square feet. What is the length of segment KE? Explain your answer or show your work. What is the length of segment JK? Explain your answer or show your work. Part C What is the length of segment HJ? Explain your answer or show your work. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (26 of 28 )

27 Part D Segments HJ and AE are parallel. What is the distance between them? Explain your answer or show your work. 47. Aisha used the three square tiles shown to model the converse of the Pythagorean Theorem. She used a side length from each of the square tiles to form a right triangle. Draw a diagram showing how to arrange the square tiles to create a right triangle. Label each square tile with the appropriate area. Using the areas in, write the equation that shows how the sum of the squares of two legs lengths equals the square of the hypotenuse length. Explain the equation you wrote. Part C Describe a method for approximating the length of the hypotenuse to the tenths place. Explain your work. Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (27 of 28 )

28 Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (28 of 28 )