10-1 10-1 Areas of Parallelograms 10-1 and Triangles 1. Plan

Objectives What You’ll Learn Check Skills You’ll Need GO for Help Lesson 1-9 1 To find the area of a parallelogram • To find the area of a Find the area of each figure. 1. 25 cm2 2. 28 in.2 4. 3 ft2 parallelogram 2 2 To find the area of a triangle 1. a square with 5-cm sides 2. a rectangle with base 4 in. and height 7 in. • To find the area of a 1 Examples triangle 3. a 4.6 m-by-2.5 m rectangle 4. a rectangle with length 3 ft and width 2 ft 2 1 Finding the Area of a 11.5 m . . . And Why Each rectangle is divided into two congruent triangles. Find the area of Parallelogram each triangle. 2 2 Finding a Missing Dimension To find the force of wind 8 units against the side of a building, 5. 6 units2 6. 2 units2 7. 3 Finding the Area of a Triangle as in Example 4 4 Real-World Connection

Math Background New Vocabulary • base of a parallelogram • altitude of a parallelogram • height of a parallelogram • base of a triangle The area of, or number of square • height of a triangle units covered by, a quadrilateral, can be described algebraically as the product of its base and height. You can demonstrate this by trans- forming, cutting, and pasting 1 Area of a Parallelogram sections of the quadrilateral to form a rectangle. Base and height The diagrams at the top of page 532 show that a parallelogram with the same base also appear in the formulas for and height as a rectangle has the same area as the rectangle. the area of a triangle (half of a parallelogram) and the lateral surface areas of prisms and Key Concepts Theorem 10-1 Area of a Rectangle pyramids. Because of this, the area of a rectangle is often given The area of a rectangle is the product of its h in terms of base and height, not base and height. length and width. A = bh b

More Math Background: p. 530C Theorem 10-2 Area of a Parallelogram The area of a parallelogram is the product of Lesson Planning and a base and the corresponding height. h Resources A = bh b

See p. 530E for a list of the resources that support this lesson. A base of a parallelogram is any of its sides. The corresponding altitude is a Vocabulary Tip PowerPoint segment perpendicular to the line containing that base, drawn from the side The term base is used to opposite the base. The height is the length of an altitude. Bell Ringer Practice represent both a segment and its length. Check Skills You’ll Need Altitude For intervention, direct students to: Base Finding the Area of a Rectangle Lesson 1-9: Example 4 Extra Skills, Word Problems, Proof 534 Chapter 10 Area Practice, Ch. 1

Special Needs L1 Below Level L2 Students may not be familiar with labeling the sides After students find the area in Example 2 of a rectangle as height and base. Point out that a algebraically, have them count the number of squares rectangle is a parallelogram with four right angles, and parts of squares and compare their answers with and the height of a rectangle is always equal to 15 square units. a side. 534 learning style: verbal learning style: visual 1 EXAMPLE Finding the Area of a Parallelogram 2. Teach nline Find the area of each parallelogram. a. b. Guided Instruction 4.5 in. 4 in. 4.6 cm 3.5 cm 1 EXAMPLE Teaching Tip 2 cm 5 in. Make sure that students under- You are given each height. Choose the corresponding side to use as the base. stand that 5 in. is the measure Visit: PHSchool.com A = bh A = bh of the entire base in part a. Web Code: aue-0775 = 5(4) = 20 Substitute. = 2(3.5) = 7 2 EXAMPLE The area is 20 in.2. The area is 7 cm2. Error Prevention If students think the base that Quick Check 2 1 Find the area of a parallelogram with base 12 m and height 9 m. 108 m corresponds to height CF is AF rather than AD, remind them that the base is always a side of the 2 EXAMPLE Finding a Missing Dimension parallelogram.

For $ABCD, find CF to the nearest tenth. F PowerPoint $ First, find the area of ABCD. Then use the D C area formula a second time to find CF. Additional Examples A = bh 13 in. 1 Find the area of the 12 in. = 10(12) = 120 Use base AB and height DE. parallelogram. The area of $ABCD is 120 in.2. For: Area Activity 10.5 m Use: Interactive Textbook, 9-1 A = bh A E B 8 m 10 in. 120 = 13(CF) Use base AD and height CF. 120 CF = 13 < 9.2 12 m CF is about 9.2 in. 96 m2

Quick Check 2 A parallelogram has sides 15 cm and 18 cm. The height corresponding to a 15-cm 2 A parallelogram has 9-in. base is 9 cm. Find the height corresponding to an 18-cm base. 7.5 cm and 18-in. sides. The height corresponding to the 9-in. base is 15 in. Find the height corresponding to the 18-in. base. 12 Area of a Triangle 7.5 in.

You can rotate a triangle about the midpoint of a side to form a parallelogram.

M h h

b b The area of the triangle is half the area of the parallelogram.

Key Concepts Theorem 10-3 Area of a Triangle The area of a triangle is half the product of a base and the corresponding height. h = 1 A 2 bh b

Lesson 10-1 Areas of Parallelograms and Triangles 535

Advanced Learners L4 English Language Learners ELL After Examples 1 and 2, have students explore Have students repeat aloud the formulas for the area possible areas for a parallelogram with side lengths of a rectangle, parallelogram, and triangle. Note that 12 cm and 10 cm. They should justify their conclusions. in “area equals one-half base times height,” the base and height must be perpendicular segments.

learning style: verbal learning style: verbal 535 Guided Instruction Abase of a triangle is any of its sides. The corresponding height is the length of the altitude to the line containing that base. Tactile Learners Have students cut out two copies 3 EXAMPLE Finding the Area of a Triangle of a triangle and then join the triangles in three ways to form Find the area of the triangle. parallelograms. Students can 1 A = bh 6.4 ft readily see that the area of each 2 1 parallelogram is twice the area = (10)(6.4) = 32 Substitute and simplify. 2 10 ft 4 ft of one triangle. The area of the triangle is 32 ft2. Connection 4 EXAMPLE Quick Check 3 Find the area of the triangle. 13 cm to Algebra 30 cm2 5 cm Have students carefully examine 12 cm the formula. Ask: What happens to force F if surface area A doubles? EXAMPLE Real-World Connection Force doubles. What happens to 4 force F if wind velocity v doubles? Structural Design When designing a building, you must be sure that the building Force quadruples. can withstand hurricane-force winds, which have a velocity of 73 mi/h or more. The 2 PowerPoint formula F = 0.004Av gives the force F in pounds exerted by a wind blowing Additional Examples against a flat surface. A is the area of the surface in square feet, and v is the wind velocity in miles per hour. 3 Find the area of XYZ. How much force is exerted by a 73 mi/h wind blowing 6 ft X directly against the side of the building shown here? 31 cm Find the area of the side of the building. 12 ft 13 cm = 1 = 1 = 2 triangle area 2 bh 2(20)6 60 ft Z Y 30 cm rectangle area = bh = 20(12) = 240 ft2 20 ft Real-World Connection 195 cm2 area of the side = 60 + 240 = 300 ft2 In 1992 this building in Use the area of the side of the building and the 4 The front of a garage is a Homestead, Florida, succumbed to the 145 mi/h velocity of the wind to find the force. square 15 ft on each side with a winds of Hurricane Andrew. 2 triangular roof above the square. F = 0.004Av Use the formula for force. The height of the triangular roof is = 0.004(300)(73)2 Substitute 300 for A and 73 for v. 10.6 ft. To the nearest hundred, = 6394.8 how much force is exerted by an 80 mi/h wind blowing directly The force is about 6400 lb, or 3.2 tons. against the front of the garage? Quick Check Use the formula F = 0.004Av2. 4 Critical Thinking Suppose the bases of the rectangle and triangle in the building about 7800 lb above are doubled to 40 ft, but the height of each figure remains the same. How is the force of the wind against the side of the building affected? The force is doubled. Resources • Daily Notetaking Guide 10-1 EXERCISES For more exercises, see Extra Skill, Word Problem, and Proof Practice. L3 • Daily Notetaking Guide 10-1— Practice and Problem Solving Adapted Instruction L1 A Practice by Example Find the area of each parallelogram. 3.5 m 26.79 in.2 Closure 20.3 m2 for Example 1 1. 2. 3. GO (page 535) 15 cm 12 cm An isosceles trapezoid has 10-m Help 5.8 m 5.7 in. 4.7 and 20-m bases, and the height is 20 cm 8 m. Find the area. Hint: Draw a 240 cm2 6 in. diagonal. 120 m2 4 m

536 Chapter 10 Area

536 Example 2 Find the value of h for each parallelogram. 3. Practice (page 535) 4.11.2 5. 6. h 0.5 13 14 0.3 h 12 h Assignment Guide 8 0.4 18 10 0.24 1 AB1-6, 11-16, 24-28 8 1613 2 AB 7-10, 17-23, 29-36 Example 3 Find the area of each triangle. (page 536) C Challenge 37-39 7. 8. 9. 14 m2 4.5 yd 3 ft2 5.7 m 5 m 6 yd 3 ft Test Prep 40-44 4 m Mixed Review 45-55 7.5 yd 4 m 3 m 2 13.5 yd 2 ft 2 ft Homework Quick Check To check students’ understanding Example 4 10. Landscaping Taisha’s Bakery has a plan for of key skills and concepts, go over (page 536) a 50 ft-by-31 ft parking lot. The four parking Exercises 2, 10, 24, 29, 30. spaces are congruent parallelograms, the driving region is a rectangle, and the two unpaved areas 10 ft Exercises 1–3 Make sure that for flowers are congruent triangles. students also square the unit a. Find the area of the surface to be paved by 50 ft of measurement. adding the areas of the driving region and 2 10b. Find the entire area the four parking spaces. 1390 ft Exercises 7–9 To help students of the lot and b. Describe another method for finding the find corresponding heights and subtract the area area of the surface to be paved. for the flowers. bases, suggest that they rotate c. Use your method from part (b) to find their textbooks so that each – ≠ 15 ft 10c. 1550 160 the area. Then compare answers from parallelogram has a horizontal 1390 ft2 31 ft parts (a) and (b) to check your work. base and vertical height. B Apply Your Skills 11. The area of a parallelogram is 24 in.2 and the height is 6 in. Find the B Exercise 12 Ask: How do you 4 in. corresponding base. know that part of the figure is a 12. Multiple Choice What is the area of the figure 14 cm square? four right angles and all at the right? B sides are congruent 64 cm2 88 cm2 8 cm 96 cm2 112 cm2 13. An isosceles right triangle has area of 98 cm2. 8 cm Find the length of each leg. 14 cm x 2 14. Algebra In a triangle, a base and a corresponding height are in the ratio 3 : 2. The area is 108 in.2.Find the base and the corresponding height. 18 in.; 12 in. 15. The area does not 15. Technology Ki used change; the height and geometry software to GPS Guided Problem Solving L3 base AB do not change. create the figure at the *right.) She constructed C k D Enrichment L4 *AB) and a point C not on Reteaching L2 GO nline AB.Then she constructed* ) Adapted Practice L1 Homework Help line k parallel to AB PracticeName Class Date L3 Visit: PHSchool.com through point C. Next, Practice 10-1 Space Figures and Nets A B 1. Choose the nets that will fold to make a cube. Web Code: aue-1001 Ki constructed point D A. B. C. D. on line k as well as AD BD # and . She dragged point D along line k to manipulate ABD. How does the Draw a net for each figure. Label each net with its appropriate dimensions.

2. 3. 4. 1 cm # 7 cm 8 cm area of ABD change? Explain. See left. 2 cm 2 cm 16 cm 32 cm 1 cm 16. Open-Ended Using graph paper, draw an acute triangle, an obtuse triangle, and 40 cm 2 Match each three-dimensional figure with its net. a right triangle, each with area 12 units . See margin. 5. 6. 7. 8.

Lesson 10-1 Areas of Parallelograms and Triangles 537 A. B. C. D.

9. Choose the nets that will fold to make a pyramid with a square base. A. B. C. D. © Pearson Education, Inc. All rights reserved.

Use Euler’s Formula to find the missing number.

10. Faces: 5 11. Faces: 7 12. Faces: 8 Edges: 7 Edges: 9 Edges: 18 Vertices: 5 Vertices: 6 Vertices: 7

537 & 17. 15 units2 Find the area of each figure. y 4. Assess Reteach J 18. 6 units2 17. $ABJF 18. #BDJ 4 F K PowerPoint 19. 6 units2 # $ 19. DKJ 20. BDKJ 2 Lesson Quiz 20. 12 units2 $ # 21. ADKF 22. BCJ A BDC 2 x 21. 27 units 23. ADJF 21 units2 O 2 4 6 8 10 12 1. Find the area of the 22. 3 units2 parallelogram. In Exercises 24–27, (a) graph the lines and (b) find the area of the triangle enclosed by the lines. 24–27. See margin. GPS 24. y = x, x = 0, y = 7 25. y = x + 2, y = 2, x = 6 10 ft 1 3 26. y =-2 x + 3, y = 0, x =-2 27. y = 4 x - 2, y =-2, x = 4 15 ft 12 ft 28. Find the area of the yellow triangular patch in the large field in the photo at the left. It has a base of 60 yd and a height of 140 yd. 4200 yd2 2 150 ft 29. Probability Ann drew these three figures on a grid. A fly lands at random at a 2. Find the area of XYZW with point on the grid. base 4 units and height 6 units. 24 square units 3. A parallelogram has 6-cm and 8-cm sides. The height corresponding to the 8-cm base is 4.5 cm. Find the height corresponding to the 6-cm base. 6 cm a. Writing Is the fly more likely to land on one of the figures or on the blank grid? Explain. Blank grid; area is 84 units2 while figures are 36 units2. 4. Find the area of RST. b. Suppose you know the fly lands on one of the figures. Is the fly more likely to R Exercise 28 land on one figure than on another? Explain. 11 m No; the figures have the same area. 5 m Coordinate Geometry Find the area of a polygon with the given vertices. 2 2 T 60 units 28 units S 6 m 30. A(3, 9), B(8, 9), C(2, -3), D(-3, -3) 31. E(1, 1), F(4, 5), G(11, 5), H(8, 1) 15 m2 32. D(0, 0), E(2, 4), F(6, 4), G(6, 0) 33. K(-7, -2), L(-7, 6), M(1, 6), N(7, -2) 5. A rectangular flag is divided 20 units2 88 units2 into four regions by its Find the area of each figure. diagonals. Two of the regions are shaded. Find the total area 34.25 ft 35.15 cm 36. 200 m of the shaded regions. 21 cm 22 in. 25 ft 120 m 40 m 17 in. 25 ft 20 cm 60 m 312.5 ft2 525 cm2 12,800 m2 C Challenge History The ancient Greek mathematician Heron is most famous for this formula for the area of a triangle in terms of the lengths of its sides a, b, and c. 187 in.2 1 A = "s(s 2 a)(s 2 b)(s 2 c), where s = 2(a + b + c) Alternative Assessment Use Heron’s Formula and a calculator to find the area of each triangle. Round your answer to the nearest whole number. Have each student draw and label 37. a = 8 in., b = 9 in., c = 10 in. 34 in.2 38. a = 15 m, b = 17 m, c = 21 m 126 m2 a triangle and a parallelogram, each with an area of 40 in.2 Have 39. a. Use Heron’s Formula to find the area of this triangle. 15 in. them write a paragraph explaining b. Verify your answer to part (a) by using the 54 in.2 9 in. 1 2 how they calculated the area of formula A = 2 bh. 54 in. each figure. 12 in. 538 Chapter 10 Area

2 16. Answers may vary. 24. a y y 7 b. 25 units Sample: 6 x 0 4 y x 4 4 4 2 x 6 6 6 538 O 2 4 6 Test Prep Test Prep Resources Multiple Choice 40. The lengths of the sides of a right triangle are 10 in., 24 in., and 26 in. For additional practice with a What is the area of the triangle? B variety of test item formats: 2 2 2 2 A. 116 in. B. 120 in. C. 130 in. D. 156 in. • Standardized Test Prep, p. 593 41. What is the area of $ABCD at the right? G D C • Test-Taking Strategies, p. 588 F. 32 in.2 G. 64 in.2 • Test-Taking Strategies with H. 91.2 in.2 J. 45.6 in.2 8 in. Transparencies 11.3 in. 42. A parallelogram has adjacent sides of 176 ft and 312 ft. The altitude to the shorter side is AB8 in. 290 ft. What is the area of the parallelogram? A A. 51,040 ft2 B. 51,352 ft2 C. 54,912 ft2 D. 55,202 ft2 43. The perimeter of an equilateral triangle is 60 m. Its height is 17.3 m. What is its area? F F. 173 m2 G. 200 m2 H. 348 m2 J. 1044 m2 $ Short Response 44. a. For ABCD, explain how to D (-3, 2) y C (2, 2) determine the length of an 2 altitude drawn to base AB . b. Find the area of $ABCD. O 2 x a-b. See margin.

- - - A ( 1, 3) B (4, 3) 44. [2] a. It is the distance between y ≠ 2 and y ≠–3, so MixedMixed ReviewReview 2 – (–3) ≠ 5. b. 25 units2 Lesson 9-7 List the symmetries in each tessellation. 45–46. See back of book. for [1] incorrect explanation GO Help 45. 46. OR incorrect answer 53. D

A B Lesson 4-5 The base of the isosceles triangle is a side N of a regular pentagon PENTA. Find the measure of each angle. ET 47. &APE 108 48. &APN 72 54. 49. &PAN 72 50. &PNA 36 51. &EPN 36 52. &ANT 36 PA G H 53–55. See margin. Lesson 3-8 Use a compass and straightedge for the following constructions. * ) * ) E F AB AD AD ' AB 53. Draw a segment and label it . Construct * so) that * ) at point A. 54. Draw a segment. Label it EF . Construct a line GH so that GH 6 EF. * ) 55. X 55. Draw a segment and label it KL . Draw a point X not on* KL) . Construct a perpendicular from point X to KL (or to KL ).

K L lesson quiz, PHSchool.com, Web Code: aua-1001 Lesson 10-1 Areas of Parallelograms and Triangles 539

25. ay 26. ay 27. a y 1 3 6 2 4 1 y x 2 x 6 y x 3 4 x 2 x 2 O 13 x 2 4 y x 4 y 2 y 2 O y 0 4 x x O 2 4 539