7-1.Your friend Alonzo owns a rug manufacturing company, which is famous for its unique designs. Each rug design has an original size as well as enlargements that are exactly the same shape. Find the area and perimeter of any enlargement of the original rug above. Your work must include the following: Diagrams for the rugs of the next two sizes (Figures 4 and 5) following the pattern shown in Figures 1, 2, and 3. Include area and perimeter of each. A description of Figure 20. What will it look like? What are its area and perimeter? Explain how you got the area and perimeter. Do not draw #20. A table and equation representing the perimeter of the rug design. A table and equation representing the area of the rug design. Figure # 1 2 3 4 5 20 n Perimeter Area 7-2. Examine the rectangle below. If the perimeter is 78 cm, find the dimensions of the rectangle. Show all your work. Verify that the area of this rectangle is 360 sq. cm. Explain how you know this.
7-3. Assume that all angles in the diagram below are right angles and that all the measurements are in centimeters. Find the area and perimeter of the figure. 7-4. Find the area of each shape. If a shape has shading, then find the area of the shaded region. Show all your work.
7-5. Kenisha thinks that the rectangle and parallelogram below have the same area. Her teammate Shaundra disagrees. Who is correct? Justify your conclusion. 7-6. Find the area of each parallelogram. 7-7. Shaundra claims that the area of a parallelogram can be found by using triangles. Do you agree? Copy the parallelogram below onto your paper. Then divide it into two triangles. Use what you know about calculating the area of a triangle to find the area of the parallelogram. It may help to draw each triangle separately so that you can rotate them and label any lengths that you know.
7-8. How do you know which dimensions to use when finding the area of a triangle? Copy each triangle below onto your paper. Then find the area of each triangle. Turning the triangles may help you discover a way to find their areas. Which numbers from each triangle did you use to find the area? For instance, in the center triangle, you probably used only the 6.8" and 5.9". Write an explanation and/or draw a diagram that would help another student understand how to choose which lengths to use when calculating the area. 7-9. Mario, Raquel, and Jocelyn are arguing about where the height is for the triangle below. The three have written their names along the part they think should be the height. Determine which person is correct. Explain why the one you chose is correct and why the other two are incorrect. 7-10. The shaded triangle below is surrounded by a rectangle. Find the area of the triangle.
7-11. Shaundra noticed that two identical trapezoids could be arranged to form a parallelogram. Is she correct? Copy the trapezoid shown below onto your paper. Carefully trace your trapezoid on another paper and cut it out with scissors. Be sure to label its bases and height as shown in the diagram. Move and rearrange the cut out trapezoid so that the two trapezoids form a parallelogram. Since you built a parallelogram from two trapezoids, you can use what you know about finding the area of a parallelogram to find the area of the trapezoid. If the bases of each trapezoid are b 1 and b 2 and the height of each is h, then find the area of the parallelogram. Then use this area to find the area of the original trapezoid. 7-12. Calculate the exact areas of the shapes below.
7-13. Find the areas of the trapezoids below. Show all your work. 7-14. Calculate the area of the shaded region below. Explain all your steps. 7-15. Copy each trapezoid below on your paper. Then find its area and perimeter. (Note: The diagrams are not drawn to scale.) 7-16. Find the area and perimeter of each shape below. Show all work.
7-17. Use what you know about the area and circumference of circles to answer the questions below. Show all work. Leave answers in terms of π. a. If the radius of a circle is 14 units, what is its circumference? What is its area? b. If a circle has diameter 10 units, what is its circumference? What is its area? c. If a circle has circumference 100π units, what is its area? d. If a circle has circumference C, what is its area in terms of C? 7-18. The city of Denver wants you to help build a dog park. The design of the park is a rectangle with two semicircular ends. (Note: A semicircle is half of a circle.) a. The entire dog park needs to be covered with grass. If grass is sold by the square foot, how much grass should you order? b. The park also needs a fence for its perimeter. A sturdy chain-linked fence costs $8 per foot. How much will a fence for the entire dog park cost? 7-19. Find the area of each figure below. Show all work. Remember to include units in your answer. 7-20. Examine the diagrams below. For each one, use geometric relationships to solve for desired information.
7-21. The giant sequoia trees in California are famous for their immense size and old age. Some of the trees are more than 2500 years old and tourists and naturalists often visit to admire their size and beauty. In some cases, you can even drive a car through the base of a tree! One of these trees, the General Sherman tree in Sequoia National Park, is the largest living thing on the earth. The tree is so gigantic, in fact, that the base has a circumference of 102.6 feet! Assuming that the base of the tree is circular, how wide is the base of the tree? That is, what is its diameter? How does that diameter compare with the length and width of your classroom? 7-22. Find the perimeter and area of each figure. 7-23. A running track design is composed of two half circles connected by two straight-line segments. Garrett is jogging on the inner lane (with radius r) while Devin is jogging on the outer (with radius R). If r = 30 meters and R = 33 meters, how much longer does Devin have to run to complete one lap?
7-24.To celebrate their victory, the girls soccer team went out for pizza. a. The goalie ate half of a pizza that had a diameter of 20 inches. What was the area of pizza that she ate? What was the length of crust that she ate? Leave your answers in exact form. That is, do not convert your answer to decimal form. b. Sonya chose a slice from another pizza that had a diameter of 16 inches. If her slice had a central angle of 45, what is the area of this slice? What is the length of its crust? Show how you got your answers. c. As the evening drew to a close, Sonya noticed that there was only one slice of the goalie s pizza remaining. She measured the central angle and found out that it was 72. What is the area of the remaining slice? What is the length of its crust? Show how you got your answer. d. A portion of a circle (like the crust of a slice of pizza) is called an arc. This is a set of connected points a fixed distance from a central point. The length of an arc is a part of the circle s circumference. If a circle has a radius of 6 cm, find the length of an arc with a central angle of 30. e. A region that resembles a slice of pizza is called a sector. It is formed by two radii of a central angle and the arc between their endpoints on the circle. If a circle has radius 10 feet, find the area of a sector with a central angle of 20.
7-25. A certain car s windshield wiper clears a portion of a sector as shown shaded below. If the angle the wiper pivots during each swing is 120, find the area of the windshield that is wiped during each swing. 7-26. Candice missed the lesson about finding the area of the triangle. Not knowing where to start, she drew a triangle and measured its sides, as shown below. After drawing her triangle, Candice said, Well, I ve measured all of the sides. I must be ready to find the area! If you think she is correct, write a description of how to use the side lengths to find the area. If you think she needs to measure anything else, copy the figure on your paper and add a line segment to represent a measurement she needs. 7-27. Use the relationships in each diagram below to solve for the given variables.
7-28. Zoe the goat is tied by a rope to one corner of a 15 meter-by-25 meter rectangular barn in the middle of a large, grassy field. Over what area of the field can Zoe graze if the rope is: (Draw diagrams for each part, show all your work.) a. 10 meters long? grazing area r = 10 m b. 15 meters long? c. 20 meters long? barn 7-29. Find the area of each. Which shape below has the least area? a. A circle with radius 5 units. b. A square with side length 9 units. c. A trapezoid with bases of length 8 and 10 units and height of 9 units. d. A rhombus with side length 9 units and height of 8 units. 7-30. For each relationship below, write and solve an equation for x. Justify your method. 7-31. Use all your circle relationships to solve for the variables in each of the diagrams below. a. The area of is 36π sq. units. b. The area of is 25π sq. units
7-32. Answer the questions below. Show all your work. a. Find the area of a circle with radius 10 units. b. Find the circumference of a circle with diameter 7 units. c. If the area of a circle is 121π square units, what is its diameter? d. If the circumference of a circle is 20π units, what is its area? 7-33. The area of the trapezoid below is 56 un 2. What is h? Show all work. 7-34. In the figure below, find the interior height (h) of the obtuse triangle. Show all work. 7-35. Use the relationships in the diagrams below to find the values of the variables, if possible. The diagrams are not drawn to scale.
7-36. Beth needs to fertilize her flowerbed, which is in the shape of a regular pentagon. A bag of fertilizer states that it can fertilize up to 150 square feet, but Beth is not sure how many bags of fertilizer she should buy. Beth does know that each side of the pentagon is 15 feet long. Copy the diagram of the regular pentagon below onto your paper. Find the area of the flowerbed and tell Beth how many bags of fertilizer to buy. Explain how you found your answer. 7-37. Recently, your school ordered a stained-glass window with the design of the school s mascot. Your student body has decided that the shape of the window will be a regular octagon, shown below. To fit in the space, the window must have a radius of 2 feet. The radius of a regular polygon is the distance from the center to each vertex. a. A major part of the cost of the window is the amount of glass used to make it. The more glass used, the more expensive the window. Your principal has turned to your class to determine how much glass the window will need. Copy the diagram onto your paper and find its area. Explain how you found your answer. b. The edge of the window will have a polished brass trim. Each foot of trim will cost $48.99. How much will the trim cost? Show all work.
7-38. Mr. Singer has a dining table in the shape of a regular hexagon. While he loves this design, he has trouble finding tablecloths to cover it. He has decided to make his own tablecloth! In order for his tablecloth to drape over each edge, he will add a rectangular piece along each side of the regular hexagon as shown in the diagram below. Using the dimensions given in the diagram, find the total area of the cloth Mr. Singer will need. 7-39. Christie has tied a string that is 24 cm long into a closed loop, like the one below. a. She decided to form an equilateral triangle with her string. What is the area of the triangle? b. She then forms a square with the same loop of string. What is the area of the square? Is it more or less than the equilateral triangle she created in part (a)? c. If she forms a regular hexagon with her string, what would be its area? Compare this area with the areas of the square and equilateral triangle from parts (a) and (b). d. What shape should Christie form to enclose the greatest area? 7-40. The Isoperimetric Theorem states that of all closed figures on a flat surface with the same perimeter, the circle has the greatest area. Use this fact to answer the questions below. a. What is the greatest area that can be enclosed by a loop of string that is 24 cm long? b. What is the greatest area that can be enclosed by a loop of string that is 18π cm long?
7-41. Below is a scale drawing of the floor plan for Nina s dollhouse. The actual dimensions of the dollhouse are 5 times the measurements provided in the floor plan below. a. Use the measurements provided in the diagram to find the area and perimeter of her floor plan. b. Draw a similar figure on your paper. Label the sides with the actual measurements of Nina s dollhouse. What is the perimeter and area of the floor of her actual dollhouse? Show all work. c. Find the ratio of the perimeters of the two figures. What do you notice? d. Find the ratio of the areas of the two figures. How does the ratio of the areas seem to be related to the scale factor? 7-42. Kelly s shape below has an area of 17 mm 2. If she enlarges the shape with a scale factor of 5, what will be the area of the enlargement? Show how you got your answer. 7-43. Examine the two similar shapes below. What is the scale factor? What is the area of the smaller figure?
7-44. Rectangle ABCD at right is divided into nine smaller congruent rectangles. Is the shaded rectangle similar to ABCD? If so, what is the scale factor? And what is the ratio of the areas? If the shaded rectangle is not similar to ABCD, explain how you know. 7-45. While ordering carpet for his rectangular office, Trinh was told by the salesperson that a 16' by 24' piece of carpet costs $800. Trinh then realized that he read his measurements wrong and that his office is actually 8' by 12'. Oh, that s no problem, said the salesperson. That is half the size and will cost $400 instead. Is that fair? Decide what the price should be. 7-46. If the side length of a hexagon triples, how does the area increase? First make a prediction. Then confirm your prediction by calculating and comparing the areas of the two hexagons shown below. 7-47. Your teacher enlarged the figure below so that the area of the similar shape is 900 square cm. What is the perimeter of the enlarged figure? Explain your method.
7-1. Examine the shape below. Extra Problems Unit 7 Find the area and perimeter of the shape. On graph paper, enlarge the figure so that the linear scale factor is 3. Find the area and perimeter of the new shape. What is the ratio of the perimeters of both shapes? What is the ratio of the areas? 7-2. While Jessie examines the two figures below, she wonders if they are similar. Decide with your team if there is enough information to determine if the shapes are similar. Justify your conclusion. 7-3. Assume Figure A and Figure B, below, are similar. If the ratio of similarity is then what is the ratio of the perimeters of A and B?
If the perimeter of Figure A is p and the linear scale factor is r, what is the perimeter of Figure B? If the area of Figure A is a and the linear scale factor is r, what is the area of Figure B? 7-4. Find the area of the shaded region for the regular pentagon below if the length of each side of the pentagon is 10 units. Assume that point C is the center of the pentagon. 7-5. The circle below is inscribed in a regular hexagon. Find the area of the shaded region. 7-6. Use the relationships in the diagrams below to solve for the given variable. Justify your solution with a definition or theorem. a.
b. The perimeter of the quadrilateral below is 202 units. c. CARD is a rhombus. d.