A C E Applications Connections Extensions Applications Investigation 3 For Exercises 1 7, find the area and perimeter of each parallelogram. Explain how you found your answers for parallelograms 2, 6, and 7. 1
8. On the grid is a family of parallelograms. a. Find the base, height, and area of each of the parallelograms. b. What patterns do you see? c. Why do you think these parallelograms belong to a family of parallelograms? 9. a. The base of a parallelogram is 1 9 2 centimeters and its height is 1 11 2 centimeters. What is the area of the parallelogram? b. The base of a rectangle is 1 9 2 centimeters and its height is 1 11 2 centimeters. What is the area of the rectangle? c. Do the parallelogram in part (a) and the rectangle in part (b) have the same perimeter? Explain. For Exercises 10 13, find the area and perimeter of each figure. 10. 11. 12. 13. 2
For Exercises 14 19, make the measurements (in centimeters) that you need to find the area and perimeter of each shape. Write your measurements on a sketch of each figure. Then, find the area and perimeter of each shape. 14. 15. 16. 17. 18. 19. 3
20. Denzel decides the shape of Tennessee is approximately that of a parallelogram, as shown below. a. Use the distances on the map to estimate the area of Tennessee. b. Suppose the actual area of Tennessee is 42,144 square miles. How does your estimate compare to the actual area? Explain. 21. Explain why these three parallelograms have the same area. For Exercises 22 27, follow the steps below. a. Sketch the described parallelogram. b. Label its base and height. c. Explain whether or not you can draw more than one parallelogram that will meet the given conditions. 22. The base is 8 centimeters and the perimeter is 28 centimeters. 23. The base is 4.5 centimeters and the area is 27 square centimeters. 24. The parallelogram is nonrectangular with a base of 10 centimeters and a height of 8 centimeters. 25. The base is 6 centimeters and the area is 30 square centimeters. 26. The area is 24 square centimeters. 27. The perimeter is 24 centimeters. 4
28. a. An equilateral triangle can be divided into equal-size triangles using line segments parallel to the opposite sides. Each segment connects two midpoints. How many parallelograms can you find in the figure? b. Are the parallelograms you found the same size? c. Suppose the area of the large triangle is 16 square units. What is the area of one of the parallelograms? 29. The Akland Middle School plans to construct a flowerbed in front of the Administration Building. The plan involves one main parallelogram surrounded by four small parallelograms, as shown. a. Find the area of one of the small parallelograms. b. Find the area of the main parallelogram. 30. Mr. Lee wants to install ceiling tiles in his recreation room. The room measures 24 feet by 18 feet. Each ceiling tile is 2 feet by 3 feet. How many ceiling tiles will he need? 31. The Lopez family bought a plot of land in the shape of a parallelogram. It is 100 feet wide (across the front) and 200 feet deep (the height). Their house covers 2,250 square feet of land. How much land is left for grass? 5
32. The Luis Park District set aside a rectangular section of land to make a park. After talking with students, the park district decided to make an area for skateboarding, an area with playground equipment, and an area with a basketball court, as shown. a. The skateboarding area takes up 2 3 of the length and 2 3 of the width of the park. What fraction of the area of the park does the skateboarding area occupy? b. The basketball court is 35 feet by 60 feet. Use this information and what you know about the skateboarding area to find the area and the perimeter of the playground area. 33. Quilters use shapes such as triangles, squares, rectangles, and parallelograms when designing quilts. This is a pattern of a 10 inch-by-10 inch quilt square on inch grid paper. a. Each parallelogram in the quilt is made from how many square inches of fabric? b. How many square inches of fabric are needed to make the five squares in the quilt square? c. The squares and the parallelograms will be sewn onto gray fabric. How many square inches of the gray fabric will be visible? 6
34. The coordinate grid at the right shows four polygons. a. Give the coordinates of all vertices of each polygon. b. Use the coordinates to find the lengths of as many sides (horizontal and vertical) as you can. c. Describe as precisely as possible each type of triangle or quadrilateral shown. 35. The coordinate grid at the right shows four polygons. a. Give the coordinates of all vertices of each polygon. b. Use the coordinates to find the lengths of as many sides as you can. c. Describe as precisely as possible each type of triangle or quadrilateral shown. 36. Don made a puzzle. He listed points that would make a polygon on a grid, but he left out some coordinates. Find the missing coordinates. a. A square with vertices A(x, 2), B(5, 6), C(1, 6), and D(m, n). b. A right triangle with a right angle at P(1, 3) and vertices Q(1, 7) and R(5, y). c. A rectangle with vertices E(3, 9), F(7, 9), G(7, 4), and H(x, y). d. A parallelogram with vertices J(3, 2), K(5, 1), L(5, 11), and M(x, y). 37. Use the polygons from Exercise 36. a. Find the lengths of the horizontal and vertical sides of each polygon. b. Find the area of each polygon. 7
38. a. Find the area of each figure. b. Design another figure that has twice the area of the following figure. Connections 39. Multiple Choice Which set of numbers is ordered from greatest to least? A. 0.215, 0.23, 2.3, 2 B. 2, 0.215, 0.23, 2.3 3 3 C. 2 3, 0.23, 0.215, 2.3 D. 2.3, 2, 0.23, 0.215 3 8
40. Multiple Choice Two quadrilaterals are congruent. Which statement is correct? F. They have the same area but different perimeters. G. They have the same perimeters but different areas. H. They have different perimeters and different areas. J. They have the same area and the same perimeter. 41. Rectangles made from Polystrips can easily tilt out of shape into parallelograms. a. Suppose a rectangle made of Polystrips tilts out of shape with the sides staying the same length. How will the angles, area, and perimeter of the new figure compare to the original? b. What relationships among the sides and angles of rectangles are also true of parallelograms? 42. Give two examples of a pair of congruent quadrilaterals that you have seen in real life. 43. Rapid City is having its annual citywide celebration. The city wants to rent a bumper-car ride. The pieces used to make the floor are 4 foot-by-5 foot rectangles. The ride covers a rectangular space that is 40 feet by 120 feet. a. How many rectangular floor pieces are needed? b. How much would it cost Rapid City to rent the floor and the bumper cars? (You will need to decide how many bumper cars are appropriate.) 9
Extensions 44. You saw earlier that for some parallelograms and triangles, the height may be outside the figure being measured. a. Sketch an example of a parallelogram with the height outside the parallelogram. Explain why the area of the parallelogram can still be calculated by multiplying the base times the height. b. Sketch an example of a triangle with the height outside the triangle. Explain why the area of the triangle can still be calculated by multiplying 1 2 times the base times the height. 45. Vlasy and Anastasia are trying to think of ways to find the area of the parallelogram below. Vlasy s Method Anastasia s Method a. Are Vlasy s and Anastasia s methods correct? Explain why they are correct or not correct. b. Compare these strategies to those you developed in class to find the area of a parallelogram. c. Will these methods work for any parallelogram? Explain. 10
46. A trapezoid is a quadrilateral with exactly one pair of parallel sides. Use these six trapezoids. Make a table to summarize what you find in parts (a) and (c). a. Find the area of each trapezoid. b. Describe how you can find the area without counting each individual square. Write a formula if possible. c. Find the perimeter of each trapezoid. d. Summarize your method for part (c) with a rule or a description. 47. Find the area and perimeter of the figure. 11
48. Auntie Judi promised to make a patchwork quilt for Amy. Amy wanted a snowflake quilt made from the block pattern below. Auntie Judi told Amy that she needed twelve patchwork blocks of this design and asked her to buy enough fabric to make it. Each block is square, 20 centimeters by 20 centimeters. a. How much yellow fabric does Amy need for one patchwork block? b. How much green fabric does she need for one patchwork block? c. What is the area of the entire snowflake quilt? d. In the whole quilt, what is the total area of the yellow fabric? The total area of green fabric? 12