Areas of Parallelograms and Triangles LESSON 11–1 Areas of Parallelograms and Triangles

Five-Minute Check (over Chapter 10) TEKS Then/Now New Vocabulary Postulate 11.1: Area Addition Postulate Key Concept: Area of a Parallelogram Example 1: Perimeter and Area of a Parallelogram Example 2: Area of a Parallelogram Postulate 11.2: Area Congruence Postulate Key Concept: Area of a Triangle Example 3: Real-World Example: Perimeter and Area of a Triangle Example 4: Use Area to Find Missing Measures Lesson Menu

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

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

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

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

Write an equation of the circle with center at (–3, 2) and a diameter of 6. 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 5-Minute Check 5

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

Mathematical Processes G.1(B), G.1(E) Targeted TEKS G.10(B) Determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change. Mathematical Processes G.1(B), G.1(E) TEKS

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

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

Concept 1

Concept 2

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

Perimeter = RS + ST + UT + RU = 32 + 20 + 32 + 20 = 104 in. Perimeter and Area of a Parallelogram Perimeter = RS + ST + UT + RU = 32 + 20 + 32 + 20 = 104 in. 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. c2 = a2 + b2 Pythagorean Theorem 202 = 122 + b2 c = 20 and a = 12 400 = 144 + b2 Simplify. Example 1

256 = b2 Subtract 144 from each side. Perimeter and Area of a Parallelogram 256 = b2 Subtract 144 from each side. 16 = b Take the positive square root of each side. The height is 16 in. UT is the base, which measures 32 in. A = bh Area of parallelogram = (32)(16) or 512 in2 b = 32 and h = 16 Answer: The perimeter is 104 in. and the area is 512 in2. Example 1

A. Find the perimeter and area of A. 88 m; 255 m2 B. 88 m; 405 m2 C. 88 m; 459 m2 D. 96 m; 459 m2 Example 1

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

Substitute 9 for the measure of the hypotenuse. 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 . ≈ 6.36 Simplify. Example 2

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

Find the area of A. 156 cm2 B. 135.76 cm2 C. 192 cm2 D. 271.53 cm2 Example 2

Concept 3

Concept 4

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

Step 1 Find the perimeter of the sandbox. Perimeter and Area of a Triangle Step 1 Find the perimeter of the sandbox. Perimeter = 16 + 12 + 7.5 or 35.5 ft Step 2 Find the area of the sandbox. Area of a triangle b = 12 and h = 9 Example 3

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

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

A. 12 boards and 14 bags of mulch B. 11 boards and 13 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? 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 Example 3

Step 1 Write an expression to represent each measure. 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 1 Write an expression to represent each measure. Let b represent the base of the triangle. Then the height is b + 7. Step 2 Use the formula for the area of a triangle to find b. Area of a triangle Example 4

120 = (b)(b + 7) Multiply each side by 2. Use Area to Find Missing Measures Substitution 120 = (b)(b + 7) Multiply each side by 2. 120 = b2 + 7b Distributive Property 0 = b2 + 7b – 120 Subtract 120 from each side. 0 = (b – 8)(b + 15) Factor. b – 8 = 0 and b + 15 = 0 Zero Product Property b = 8 b = –15 Solve for b. Example 4

Step 3 Use the expressions from Step 1 to find each measure. Use Area to Find Missing Measures Step 3 Use the expressions from Step 1 to find each measure. Since a length cannot be negative, the base measures 8 inches and the height measures 8 + 7 or 15 inches. Answer: b = 8 in., h = 15 in. Example 4

A. base = 56 in. and height = 10 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. 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. Example 4

Areas of Parallelograms and Triangles LESSON 11–1 Areas of Parallelograms and Triangles