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.