Areas of Tropezoids, Rhombuses, and Kites


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1 102 Areas of Tropezoids, Rhombuses, and Kites MathemaHcs Florida Standards MAFS.912.GMG.1.1 Use geometric shapes, their measures, and their properties to describe objects. MP1. MP3, MP 4,MP6 Objective To find the area of a trapezoid, rhombus, or kite Getting Ready! Rearranging figures into familiar shapes is an example of the Solve a Simpler Problem strategy. Drow o tropezotd on a sheet of graph poper. Label the bases bj and bg. Draw its midsegment. Cut out the trapezoid. and then cut it along the midsegment. Rotate the top part of the trapezoid 180 so that bi ond bg now form one long base. How can you use this new figure to find the area of the trapezoid? Explain your reasoning. PRA^Icfe^S Understanding You can find the area of a trapezoid when you know its height and the lengths of its bases. ^ Lesson Vocabulary height of a trapezoid The height of a trapezoid is the perpendicular distance between the bases. Theorem 104 Area of o Trapezoid The area of a trapezoid is half the product of the height and the sum of the bases. A = ^h{bi + b2) Which borders of Nevada can you use as the bases of a trapezoid? The two parallel sides of Nevada form the bases of a trapezoid. Problem 1 Area of a Trapezoid Geography What is the approximate area of Nevada? A = + ^2) Use the formula for area of a trapezoid. = (309)( ) Substitute 309 for h, 205 for bi, and 511 for 62 = 110,622 Simplify. The area of Nevada is about 110,600 mi^. Gotit? 1. What is the area of a trapezoid with height * 7 cm and bases 12 cm and 15 cm? 205 mi 309 mi (s) Reno ^ Carson City 511 ml C PowerGeometry.com Lesson 102 Areas of Trapezoids, Rhombuses, and Kites 623
2 Problem 2 Finding Area Using a Right Triangle How are the sides related in a triangle? The length of the hypotenuse Is 2 times the length of the shorter leg, and the longer leg Is V3 times the length of the shorter leg. What is the area of trapezoid PQRSl You can draw an altitude that divides the trapezoid into a rectangle and a triangle. Since the opposite sides of a rectangle are congruent, the longer base of the trapezoid is divided into segments of lengths 2 m and 5 m. h = 2V3 A = + fj2) = (2V5)(7f5) = 12V3 longer leg = shorter leg Use the trapezoid area formula. Substitute 2\/3 for h, 1 for bi, and 5for /)2. Simplify. The area of trapezoid PQRS is 12 Vs m^. S 5m /? 7 m 5 5m ft P 2m 5m 0 Got It? 2. Reasoning In Problem 2, suppose h decreases so that map = 45 while angles R and Q and the bases stay the same. What is the area of trapezoid PQRSl Essential Understanding You can find the area of a rhombus or a kite when you know the lengths of its diagonals. Theorem 105 Area of o Rhombus or o Kite The area of a rhombus or a kite Is half the product of the lengths of its diagonals. A Rhombus Kite V Problem 3 Finding the Area of a Kite Do you need to know the side lengths of the kite to find its area? No. You only need the lengths of the diagonals. What is the area of kite KLMNl Find the lengths of the two diagonals: ii;:m = = 7m and IN = = 6 m. A = k(l\d2 Use the formula for area of a kite. = Substitute 7 for dy and 6 for di = 21 Simplify. The area of kite KLMN is 21 m^. Got It? 3. What is the area of a kite with diagonals that are 12 in. and 9 in. long? 624 Chapter 10 Area
3 TV.*:' " How can you find the length of >^? AB is a leg of right AABC. You can use the Pythagorean Theorem, a2 + /j2 _ (2^ fjpj its length. Problem 4\ Finding the Area of a Rhombus Car Pooling Ihe High OccupancyVehicle(HOV) lane is marked by a series of "diamonds " or rhombuses painted on the pavement. What is the area of the HOV lane diamond shown at the right? AABC is a right triangle. Using the Pythagorean Theorem, AB = Vb.S^ 2.5^ = 6. Since the diagonals of a rhombus bisect each other, the diagonals of the HOV lane diamond are 5 ft and 12 ft. A = hdido 2"1"2 = (5X12) = 30 Use the formula for area of a rhombus. Substitute 5 for d] and 12 for ^2. Simplify. The area of the HOV lane diamond is 30 Got It? 4. A rhombus has sides 10 cm long. ^ If the longer diagonal is 16 cm, what is the area of the rhombus? & Lesson Check Do you know HOW? Find the area of each figure. Do you UNDERSTAND? MATHEMATICAL PRACTICES 7. Vocabulary Can a trapezoid and a parallelogram with the same base and height have the same area? Explain. ' 8 n. 27 in. 33 ft 8. Reasoning Do you need to know all the side lengths to find the area of a trapezoid? Reasoning Can you find the area of a rhombus if you only know the lengths of its sides? Explain. 10. Reasoning Do you need to know the lengths of the sides to find the area of a kite? Explain. 20 m 1 cm sowerg eo metry.co m Lesson 102 Areas of Trapezoids, Rhombuses, and Kites 625
4 Practice and ProblemSolving Exercises MATHEMATICAL PRACTICES Practice Find the area of each trapezoid ^ See Problem in cm in. I 38 In. j 8.5 cm 9.7 cm 18 ft 1] Vft 6 ft 14. Find the area of a trapezoid with bases 12 cm and 18 cm and height 10 cm. 15. Find the area of a trapezoid with bases 2 ft and 3 ft and height ft. 16. Geography The border of Tennessee resembles a trapezoid with bases 340 mi and 440 mi and height 110 mi. Estimate the area of Tennessee by finding the area of the trapezoid. Find the area of each trapezoid. If your answer is not an integer, leave it in simplest radical form. ^ See Problem ft, 6 ft L 3 ft ft/ /eo" 15 ft r 19. Find the area of each kite. 20.,2 in m/ 3 3 m 4 nn\. 22. /4 ft ^ See Problem 3. 6ft\ 4ft\ Find the area of each rhombus. ^ See Problem \loin. y/o in. 26. Think About a Plan A trapezoid has two right angles, 12m and 18m bases, and an 8m height. Sketch the trapezoid and find its perimeter and area. Are the right angles consecutive or opposite angles? How does knowing the height help you find the perimeter? 626 Chapter 10 Area
5 27. Metallurgy The end of a gold bar has the shape of a trapezoid with the measurements shown. Find the area of the end. 28. OpenEnded Draw a kite. Measure the lengths of its diagonals. Find its area. Find the area of each trapezoid to the nearest tenth cm 3 cm Jl 4 cm 1 cm ft 30 1^9 ft 6.9 cm 4.4 cm 9.2cm y Coordinate Geometry Find the area of quadrilateral QRST. 33. y Ir 4 n T 2 S.1 X What is the area of the kite at the right? Ca> 90 m^ CO 135 m^ CO 108 m2 CO a. Coordinate Geometry Graph the lines r = 0, x = 6, y = 0, and y = x + 4. b. What type of quadrilateral do the lines form? c. Find the area of the quadrilateral. Find the area of each rhombus. Leave your answer in simplest radical form. 39. Z in. 40. Visualization The kite has diagonals dj and congruent to the sides of the rectangle. Explain why the area of the kite is did241. Draw a trapezoid. Label its bases bi and >2 and its height h. Then draw a diagonal of the trapezoid. a. Write equations for the area of each of the two triangles formed. b. Writing Explain how you can justify the trapezoid area formula using the areas of the two triangles. d2 C PowerGeometry.com Lesson 102 Areas of Trapezoids, Rhombuses, and Kites 627
6 challenge 42. Algebra One base of a trapezoid is twice the other. The height is the average of the two bases. The area is 324 cm^. Find the height and the bases. {Hint: Let the smaller base be x) 43. Sports Ty wants to paint one side of the skateboarding ramp he built. The ramp is 4 m wide. Its surface is modeled by the equation y = 0.25^:^. Use the trapezoids and triangles y = 0.25x2 shown to estimate the area to be painted. 44. In trapezoid ABCD at the right, AB 1 DC. Find the area of ABCD. Standardized Test Prep SAT/Aa 45. The area of a kite is 120 cm^. The length of one diagonal is 20 cm. What is the length of the other diagonal? CS5 12 cm dz) 20 cm CO 24 cm CO 48 cm 46. AABC ~ AXYZ. AB = 6,BC = 3, and CA = 7. Which of the following are NOT possible dimensions of AXYZl CE:>X1'=3, yz= 1.5,ZX=3.5 Ch:)XY= 10, YZ= 7, ZX= 11 CO xy=9, YZ=4.5,ZX'= 10.5 CL^ XY= 18. YZ = 9, ZX= 21 Short.Response 47. Draw an angle. Construct a congruent angle and its bisector. r Mixed Review 48. Find the area of a right isosceles triangle that has one leg of length 12 cm. 49. A right isosceles triangle has area ft^. Find the length of each leg. 50. Find the measure of an interior angle of a regular nonagon. Get Ready! To prepare for Lesson 103, do Exercises Find the area of each regular polygon. Leave radicals in simplest form. ^ See Lesson ^ See Lesson 61. ^ See Lesson cm ' 10 ft 628 Chapter 10 Area
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