# The height of a trapezoid is the. EXAMPLE Real-World Connection

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1 0-. lan Objectives To find the area of a trapezoid To find the area of a rhombus or a kite Examples eal-world onnection Finding rea Using a Triangle Finding the rea of a Kite 4 Finding the rea of a hombus 0- What You ll Learn To find the area of a trapezoid To find the area of a rhombus or a kite... nd Why To use a map and the trapezoid area formula to approximate the area of rkansas, as in Example reas of Trapezoids, hombuses, and Kites heck kills You ll Need GO for Help Lesson 0- Write the formula for the area of each type of figure.. a rectangle bh or <w. a triangle bh Find the area of each trapezoid by using the formulas for area of a rectangle and area of a triangle M K U T 9 units 7 units.5 units New Vocabulary height of a trapezoid Math ackground rea formulas for trapezoids, rhombuses, and kites are derived from area formulas for triangles and parallelograms, and depend on properties of quadrilaterals such as opposite sides of a parallelogram are parallel and diagonals of a rhombus and of a kite are perpendicular. rea of a Trapezoid Vocabulary Tip The term base can refer to either a line segment or its length. In Lesson 6-5, you learned that the bases of a trapezoid are the parallel sides and the legs are the nonparallel sides. The height of a trapezoid is the perpendicular distance h between the bases. Leg ase b h b ase Leg More Math ackground: p. 50 Lesson lanning and esources ee p. 50E for a list of the resources that support this lesson. ell inger ractice heck kills You ll Need For intervention, direct students to: Finding the rea of a arallelogram Lesson 0-: Example Extra kills, Word roblems, roof ractice, h. 0 Finding the rea of a Triangle Lesson 0-: Example Extra kills, Word roblems, roof ractice, h. 0 Fort mith 4 mi Key oncepts Theorem 0-4 rea of a Trapezoid 50 mi 90 mi Little ock Mammoth pring uick heck 540 hapter 0 rea The area of a trapezoid is half the product of the height and the sum of the bases. h(b + b ) eal-world onnection Geography pproximate the area of rkansas by finding the area of the trapezoid shown. h(b + b ) Use the formula for area of a trapezoid. = (4)( ) ubstitute 4 for h, 90 for b, and 50 for b. = 5,40 implify. The area of rkansas is about 5,40 mi. Find the area of a trapezoid with height 7 cm and bases cm and 5 cm cm h b b 540 pecial Needs L fter Example, have students use a map to approximate the areas of other trapezoid-shaped states such as North akota and onnecticut. Then have them research the actual areas for comparison. learning style: tactile elow Level L fter the visual proof of Theorem 0-5, have students draw and color a kite or a rhombus within a rectangle and then use paper cutting to match pairs of congruent triangles. learning style: tactile

2 E E E 4 E 5 E E Test-Taking Tip You can also find the area by adding the areas of the triangle and rectangle. uick heck roperties of special right triangles can help you find the area of a trapezoid. Finding rea Using a Triangle Multiple hoice What is the area of trapezoid? 6 m 0! m! m You can draw an altitude that divides the trapezoid into a rectangle and a triangle. ince the opposite sides of a rectangle are congruent, the longer base of the trapezoid is divided into segments of lengths m and. Find h. h =! longer leg shorter leg?! h(b + b ) Use the trapezoid area formula. = (!)(7 + 5) ubstitute. =! implify. The area of trapezoid is! m. The answer is. In Example, suppose h is made smaller so that m& = 45 while bases and angles and are unchanged. Find the area of trapezoid. m Finding reas of hombuses and Kites hombuses and kites have perpendicular diagonals. This property allows you to find areas using the following theorem. Key oncepts Theorem 0-5 rea of a hombus or a Kite The area of a rhombus or a kite is half the product of the lengths of its diagonals. d d d h m 7 m 7 m = m d. Teach Guided Instruction lternative Method The formula for the area of a trapezoid also can be written b h + b h. Have students partition a trapezoid into two triangles to illustrate the formula above, as is done in Exercise 8. Teaching Tip sk: Why do you need to divide the longer base into -m and 5-m segments? You need to know the length of the shorter leg to find h. dditional Examples car window is shaped like the trapezoid shown. Find the area of the window. 504 in. Find the area of trapezoid. ft 6 in. 0 in. ft 8 in. Finding the rea of a Kite Find the area of kite KLMN. For the two diagonals, KM = + 5 = 7 m and LN = + = 6 m. d d Use the formula for area of a kite. (7)(6) ubstitute 7 for d and 6 for d. implify. m K L N m m M 6 ft 6 ft uick heck The area of kite KLMN is m. Find the area of a kite with diagonals that are in. and 9 in. long. 54 in. Lesson 0- reas of Trapezoids, hombuses, and Kites 54 dvanced Learners L4 Have students explore whether a quadrilateral that is not a rhombus or a kite can satisfy the formula in Theorem 0-5. learning style: verbal English Language Learners ELL raw a variety of trapezoids in different orientations on the board. Have students identify bases and heights. tudents must understand that vertical or horizontal sides do not distinguish bases or heights. learning style: visual 54

3 Guided Instruction Tactile Learners For the Visual roof of the rea of a hombus or a Kite Theorem, have students draw a kite or rhombus and use paper folding to identify the diagonals. Have students carefully cut along the diagonals and then fold back the right triangles along the sides of the rhombus or kite. The right angles will be vertices of a rectangle with area twice that of the original figure. lternative Method tudents also can find the area by adding the areas of the four right triangles. 4 dditional Examples Find the area of kite XYZW. X 5 cm Find the area of rhombus TU. U 0 ft cm cm cm 4 cm Y W 4 ft ft T Z uick heck Example (page 540) 4 The fact that the diagonals of a rhombus bisect each other can help you find the area. Find the area of rhombus. Finding the rea of a hombus #E is a right triangle. Using a ythagorean triple, E = 9. ince the diagonals of a rhombus bisect each other, = 4 and = 8. d d Use the formula for area of a rhombus. (4)(8) ubstitute 4 for d and 8 for d. 6 implify. The area is 6 m. 4 ritical Thinking In Example 4, explain how you can use a ythagorean triple to conclude that E = 9. 9 ± 5 EXEIE For more exercises, see Extra kill, Word roblem, and roof ractice. ractice and roblem olving GO ractice by Example for Help nline Visit: hchool.com Web ode: aue-0775 Find the area of each trapezoid.. in.. 4. cm. 6 in. 8.5 cm 8 ft 9 ft 6 ft 9.7 cm 8 in. 47 in cm 08 ft 4. Geography pproximate the area of Nevada by finding the area of the trapezoid shown. 0,6 mi 5. Find the area of a trapezoid with bases cm and 8 cm and height 0 cm. 50 cm 6. Find the area of a trapezoid with bases 5 ft and ft and height ft. ft 6 7. Geography The border of Tennessee resembles a trapezoid with bases 4 mi and 48 mi, and height mi.pproximate the area of Tennessee by finding the area of this trapezoid. about 4,90 mi 0i Humboldt. E 09 mi eno arson ity m Las Vegas i esources aily Notetaking Guide 0- L aily Notetaking Guide 0- dapted Instruction L losure Example (page 54) Find the area of each trapezoid. If your answer is not an integer, leave it in simplest radical form ft 9. 6 m 0. ft 8 ft 0 ft 0 m 8 m 6 ft 7 m 5" ft 5 ft Use the formula for the area of a rhombus to show that the area of a square with 0-in. sides is 900 in. (0 )(0 ) " " (0)(0)() 900 in. 54 hapter 0 rea 54

4 eal-world Example (page 54) Example 4 (page 54) pply Your kills On each gold bar the four trapezoidal faces tip inwards. This simplifies the molding process. GO onnection nline Homework Help Visit: Hchool.com Web ode: aue-00 G Find the area of each kite.. in.. m. m 4 ft 8 in. 8 in. 4 m 8 in. m 80 in. 8 m Find the area of each rhombus in. 00 ft 8 in. 0 ft 0 ft 7. The end of the rain gutter has the shape of a trapezoid with the measurements shown. Find the area of this end. 0 in. 8. trapezoid has two right angles, -m and 8-m bases, and 8-m height. a c. ee margin. 4 in. a. ketch the trapezoid. b. Find the perimeter. c. Find the area. 9. Open-Ended raw a kite. Measure the lengths of its diagonals. Find its area. heck students work. Gold ars Find the area of each trapezoidal face of the gold bars. 0. End face: bases 4 cm and cm, height cm. 9 cm. ide face: bases 8 cm and 5 cm, height cm. 9.5 cm Find the area of each trapezoid to the nearest tenth... cm. 8 ft 4. 4 cm 0 cm 9 ft cm cm 96 in. 4 m 49.9 ft oordinate Geometry In Exercises 5 7, find the area of quadrilateral T. 8 units 5 units 5 units 5. y 6. y 7. y 4 x 4 - O x - - T T 4 x T 8. Multiple hoice What is the area of the kite at the right? 90 cm 08 m 5 cm 6 m 9. a. oordinate Geometry Graph the lines x = 0, x = 6, y = 0, and y = x + 4. ee margin. b. What type of quadrilateral do the lines form? trapezoid c. Find the area of the quadrilateral. 4 units.8 m.7 m m 9 m 6 ft 4 ft 4 ft 45 6 m 6 in. 4 in. 6 m. m. ractice ssignment Guide -0, 7, 8, 0-4, -6, 9, 5-, 4 hallenge 5-7 Test rep 8-4 Mixed eview 4-48 Homework uick heck To check students understanding of key skills and concepts, go over Exercises 4, 4, 0, 6,. uditory Learners Exercise 6 iscuss as a class how to write and solve the appropriate formula. Have one student give each step verbally while another student writes the steps on the board. Exercise 8 sk: What do you need to find the area of the kite? the length of its diagonals How could you find the length of the diagonals? apply properties of triangles G Enrichment Guided roblem olving eteaching dapted ractice ractice Name lass ate ractice 0- Make an isometric drawing of each cube structure.... For each figure, (a) make a foundation drawing, and (b) make an orthographic drawing pace Figures and rawings L4 L L L L lesson quiz, Hchool.com, Web ode: aua-00 Lesson 0- reas of Trapezoids, hombuses, and Kites 54 earson Education, Inc. ll rights reserved. escribe the cross section in each diagram raw and describe a cross section formed by a rectangular prism and the plane described.. a horizontal plane intersecting opposite faces of the prism 4. a vertical plane intersecting the front and right faces of the prism 54

5 lternative Method Exercise 0 hallenge students to prove that the rhombus is a square, find the length of a side, and then use the formula for the area of a square. onnection to lgebra Exercise 5 tudents must solve a quadratic equation in x. onnection to alculus Exercise 6 pproximating the area between a curve and the x-axis by using trapezoids is taught in integral calculus. Using calculus, the exact area of shapes with curved sides can be found. sk: For what curved figure do you already know an area formula? circle Exercise 7 To solve, students must draw perpendicular segments to from point and point. 4. ample: Each kite section is one-half of the corresponding rectangle section. eal-world hallenge onnection The curve of a half pipe is two quarter circles joined by a horizontal segment. x Find the area of each rhombus. Leave your answer in simplest radical form cm. " m. 45 cm 8 in. 4 m 8". raw a trapezoid. Label its bases and height b, b, and h, respectively. Then draw a diagonal of the trapezoid. b h; b h 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. ee margin. 4. Visualization The kite has diagonals d and d congruent to the sides of the rectangle. Explain why the area of the kite is d d. d 5. lgebra One base of a trapezoid is twice the other. The height is the average of the two bases. The area is 4 cm. Find the height and the bases. (Hint: Let d the smaller base be x.) b cm, b 4 cm, h 8 cm 6. Gravity ports Ty wants y to paint one end of his homemade skateboarding y 0.5x ramp. The ramp is 4 m wide. Its surface is modeled by the equation y = 0.5x. Use the trapezoids and triangles shown to estimate the area to be painted.. O x 7. In trapezoid, 6. 5 in. Find the area of. 0 in ! or about 86.6 in. 5 0 in. 0 Test rep Multiple hoice 8. The area of a kite is 0 cm. The length of one diagonal is 0 cm. What is the length of the other diagonal?. cm. 0 cm. 4 cm. 48 cm 0 m 9. What is the area of the trapezoid at the right? H F. 9 m G. 60 m 6 m H. 78 m J. 96 m 6 m 40. The lengths of the sides of a rhombus and one of its diagonals are each 0 m. What is the area of the rhombus?. 00! m. 50! m. 5! m..5! m 544 hapter 0 rea 544. a. m b. 48 m. b. The area of the c. 0 m trapezoid is the sum 8 m of the areas of the triangles, so 8 m b h ± b h h(b ± b ).

6 GO hort esponse Mixed eview for Help Lesson 0- Lesson 5-4. The area of a trapezoid is 00 ft. The sum of the two bases is 5 ft. What is the height of the trapezoid? H F. ft G. 4 ft H. 8 ft J. 0 ft 4. The area of an isosceles trapezoid is 60 cm. Its height is 8 cm and the length of one leg is 0 cm. a b. ee margin. a. raw and label a diagram representing the given information. b. Find the length of each base. how your work. 4. Find the area of an isosceles right triangle that has one leg of length cm. 7 cm 44. n isosceles right triangle has area of.5 ft. Find the length of each leg. 5 ft Fill in the blank with always, sometimes, or never to form a true statement. 45. The incenter of a triangle 9 lies inside the triangle. always 46. The orthocenter of a triangle 9 lies outside the triangle. sometimes 4. ssess & eteach Lesson uiz. Find the area of a trapezoid with bases cm and 9 cm and height 9 cm. 99 cm. Find the area of a trapezoid in a coordinate plane with vertices at (, ), (, 6), (5, 9), and (5, ). 6 square units Find the area of each figure in Exercises 5. Leave your answers in simplest radical form.. trapezoid 0 in. 47. The centroid of a triangle 9 lies on the triangle. never 4 in. Lesson Find the measure of an interior angle of a regular 9-gon [] a. 0 cm 8 cm b. In the rt. k shown, the other leg is 6 cm. ince there are O >, the residents are known more often for their foreign policy than for their mathematical creativity. James N Garfield, the 0th resident of the a United tates, is an exception. In c 876, Garfield demonstrated this proof of the ythagorean Theorem. In the diagram, NM and are congruent right M b triangles with sides of lengths a, b, ab c and c. The legs of isosceles right triangle N have length c. The c three triangles form trapezoid c MN. The sum of the areas of the three triangles equals the area of trapezoid MN. Hchool.com int in Time Lesson 0- reas of Trapezoids, hombuses, and Kites 545 longer base is x ± and the shorter base is x. 60 (8)(x ± x ± ) and x 4. ases are 4 cm and 6 cm. reas of Triangles rea of Trapezoid ab (a b)(a b) ab c a ab b a b c a b a b For: Information about ythagorean Theorem proofs Web ode: aue-0 [] correct answer without explanation O correct explanation and calculation error 94.5" in. 4. kite with diagonals 0 m and 0" m long 00" m 5. rhombus MNO 9 mm M 840 mm lternative ssessment Have students work with partners to draw and label a trapezoid, a rhombus, and a kite, each with an area of 00 cm, and write a paragraph explaining how they calculated the area of each figure. Test rep N 4 mm 60 esources For additional practice with a variety of test item formats: tandardized Test rep, p. 59 Test-Taking trategies, p. 588 Test-Taking trategies with Transparencies O 545

### Finding Angle Measures. Solve. 2.4 in. Label the diagram. Draw AE parallel to BC. Simplify. Use a calculator to find the square root. 14 in.

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