Areas of Parallelograms and Triangles LESSON 11–1

Over Chapter 10 5-Minute Check 1 Name a radius. A. B. C. D.

Over Chapter 10 5-Minute Check 2 Name a chord. A. B. C. D.

Over Chapter 10 5-Minute Check 3 Name a diameter. A. B. C. D.

Over Chapter 10 5-Minute Check 4 A.90 B.120 C.160 D.170

Over Chapter 10 5-Minute Check 5 A.(x + 3) + (y – 2) = 9 B.(x – 3) + (y + 2) = 6 C.(x + 3) 2 + (y – 2) 2 = 9 D.(x – 3) 2 + (y + 2) 2 = 6 Write an equation of the circle with center at (–3, 2) and a diameter of 6.

Over Chapter 10 5-Minute Check 6 A.chord B.diameter C.secant D.tangent Which of the following figures is always perpendicular to a radius of a circle at their intersection on the circle?

Then/Now You found areas of rectangles and squares. Find perimeters and areas of parallelograms. Find perimeters and areas of triangles.

Vocabulary base of a parallelogram height of a parallelogram base of a triangle height of a triangle

Concept 1

Concept 2

Example 1 Perimeter and Area of a Parallelogram Find the perimeter and area of PerimeterSince opposite sides of a parallelogram are congruent, RS UT and RU ST. So UT = 32 in. and ST = 20 in.

Example 1 Perimeter and Area of a Parallelogram Area Find the height of the parallelogram. The height forms a right triangle with points S and T with base 12 in. and hypotenuse 20 in. c 2 = a 2 + b 2 Pythagorean Theorem 20 2 = b 2 c = 20 and a = = b 2 Simplify. Perimeter= RS + ST + UT + RU = = 104 in.

Example 1 Perimeter and Area of a Parallelogram 256= b 2 Subtract 144 from each side. 16= bTake the positive square root of each side. A= bhArea of parallelogram = (32)(16) or 512 in 2 b = 32 and h = 16 The height is 16 in. UT is the base, which measures 32 in. Answer: The perimeter is 104 in. and the area is 512 in 2.

Example 1 A.88 m; 255 m 2 B.88 m; 405 m 2 C.88 m; 459 m 2 D.96 m; 459 m 2 A. Find the perimeter and area of

Example 2 Area of a Parallelogram Step 1Use a 45°-45°-90° triangle to find the height h of the parallelogram. Find the area of

Example 2 Area of a Parallelogram Recall that if the measure of the leg opposite the 45° angle is h, then the measure of the hypotenuse is Substitute 9 for the measure of the hypotenuse. Divide each side by. ≈ 6.36Simplify.

Example 2 Area of a Parallelogram Step 2Find the area. A = bhArea of a parallelogram ≈ (12)(6.36) b = 12 and h = 6.36 ≈ 76.3Multiply. Answer:76.3 square units

Example 2 A.156 cm 2 B cm 2 C.192 cm 2 D cm 2 Find the area of

Concept 3

Concept 4

Example 3 Perimeter and Area of a Triangle SANDBOX You need to buy enough boards to make the frame of the triangular sandbox shown and enough sand to fill it. If one board is 3 feet long and one bag of sand fills 9 square feet of the sandbox, how many boards and bags do you need to buy?

Example 3 Perimeter and Area of a Triangle Step 1Find the perimeter of the sandbox. Perimeter = or 35.5 ft Step 2Find the area of the sandbox. Area of a triangle b = 12 and h = 9

Example 3 Perimeter and Area of a Triangle Step 3Use unit analysis to determine how many of each item are needed. Boards Bags of Sand boards

Example 3 Perimeter and Area of a Triangle Answer: You will need 12 boards and 6 bags of sand. Round the number of boards up so there is enough wood.

Example 3 A.12 boards and 14 bags of mulch B.11 boards and 13 bags of mulch C.12 boards and 13 bags of mulch D.11 boards and 14 bags of mulch PLAYGROUND You need to buy enough boards to make the frame of the triangular playground shown here and enough mulch to fill it. If one board is 4 feet long and one bag of mulch covers 7 square feet, how many boards and bags do you need to buy?

Example 4 Use Area to Find Missing Measures ALGEBRA The height of a triangle is 7 inches more than its base. The area of the triangle is 60 square inches. Find the base and height. Step 1Write an expression to represent each measure. Let b represent the base of the triangle. Then the height is b + 7. Step 2Use the formula for the area of a triangle to find b. Area of a triangle

Example 4 Use Area to Find Missing Measures Substitution 120=(b)(b + 7)Multiply each side by =b 2 + 7bDistributive Property 0=b 2 + 7b – 120Subtract 120 from each side. 0=(b – 8)(b + 15)Factor. b – 8 = 0 and b + 15 = 0Zero Product Property b = 8 b = –15Solve for b.

Example 4 Use Area to Find Missing Measures Step 3Use the expressions from Step 1 to find each measure. Since a length cannot be negative, the base measures 8 inches and the height measures or 15 inches. Answer: b = 8 in., h = 15 in.

Example 4 A.base = 56 in. and height = 10 in. B.base = 28 in. and height = 40 in. C.base = 20 in. and height = 56 in. D.base = 26 in. and height = 38 in. ALGEBRA The height of a triangle is 12 inches more than its base. The area of the triangle is 560 square inches. Find the base and the height.

Areas of Parallelograms and Triangles LESSON 11–1