Developing Formulas for Triangles

**KEY** **Developing Formulas for Triangles and Quadrilaterals** **Find each measurement.** 1. A = 2xy mi² P = 2(y) + 2(2x) P = 2y + 4x mi 2. A = s² A = (a - b)² A = (a - b)(a - b) = a² - ab - ab + b² A = (a² - 2ab + b²) ft² 3. the height of a parallelogram in which A = 98 cm² and b = 8x cm A = bh 98 = 8x * h h = 98 / 8x = 12.25 / x cm 4. the area of the triangle A = 1/2 bh A = 1/2 (x)(3x) A = 1.5x² in² 5. 24² + y² = 51² 576 + y² = 2601 y² = 2025 y = 45 24² + x² = 26² 576 + x² = 676 x² = 100 x = 10 A = 1/2 bh A = 1/2 (24)(10) A = 120 in² 6. the area of a trapezoid in which b₁ = 3a km, b₂ = 6a km, and h = (10 + 4c) km A = 1/2 (b₁ + b₂)h A = 1/2 (3a + 6a)(10 + 4c) A = 4.5a(10 + 4c) A = (45a + 18ac) km² 7. (4.8)² + 9² = y² 23.04 + 81 = y² 104.04 = y² y = 10.2 A = 1/2 d₁d₂ 9.84 = 1/2 (9.6)d₂ 9.84 = 4.8d₂ d₂ = 16.4 A = 1/2 d₁d₂ A = 1/2 (10.2)(16.4) A = 83.64 yd² 8. the area of the rhombus A = 1/2 d₁d₂ A = 1/2 (x + 4)(2y - 4) A = 1/2 (2xy - 4x + 8y - 16) A = xy - 2x + 4y - 8 9. d₂ of the kite in which d₁ = (a - 4) and A = (2a² - 8a) ft² A = 1/2 d₁d₂ 2a² - 8a = 1/2 (a - 4)d₂ (4a² - 16a) / (a - 4) = (a - 4)d₂ / (a - 4) d₂ = (4a² - 16a) / (a - 4) = 4a(a - 4) / (a - 4) d₂ = 4a **KEY** **Problem Solving** **Developing Formulas for Triangles and Quadrilaterals** 1. The area of trapezoid HJKL is 385 mm². Find LM to the nearest tenth. 385 = (9 + 23)(h)/2 770 = 32h h = 24.06 mm sin(40) = 14/LM LM * sin(40) = 14 LM = 14 / sin(40) LM ≈ 21.8 mm 2. Samantha is buying stones for a 3 ft by 3 ft walkway. She needs to buy 10% extra for cutting stones for the corners and ends. The stones are rectangles 7 inches long and 4 inches wide and cost $0.97 each. About how much will   272 stones cost? 272 * 0.97 = 263.84 3. The length of a rectangular pool is 2 feet less than twice the width. If the area of the pool is 264 ft², what are the dimensions of the pool? (2x - 2)(x) = 264 2x² - 2x - 264 = 0 2(x - 12)(x + 11) = 0 x = 12 Dimensions: 22 ft by 12 ft 4. Find the area of the kite. Round to the nearest tenth. (2.3)² + y² = 4² 5.29 + y² = 16 y² = 10.71 y = 3.3 cm A = 1/2(11.5)(4.6) A = 26.5 cm² **Choose the best answer.** 5. A parallelogram has sides of length 30 centimeters and 18 centimeters. One of its angles measures   58°. Which is the best estimate for the area of the parallelogram?   sin(58) = h/18 18 * sin(58) = h h ≈ 15.2 cm Area ≈ 30 * 15.2 ≈ 457.9 cm² 6. Jamie is cutting out 32 right triangles from fabric for her quilt. The shortest side of each triangle is 2 inches, and the longest side is 5 inches. How much fabric will she use to cut out all the triangles?   2² + x² = 5² x² = 21 x = √21 Area of one triangle = 1/2 * 2 * √21 = √21 in² Total area = 32 * √21 ≈ 146.6 in² 7. In rhombus ABCD, the length of diagonal BD is 2/3 the length of diagonal AC. If the area of the figure is 75 cm², find BD. A = 1/2 (x)(2/3x) 75 = 1/3x² 225 = x² x = 15 BD = 2/3 * 15 = 10 cm 8. The house is made from 7 puzzle pieces. The pieces are triangles and parallelograms. If the area of the chimney is 1/8 in²,   what is the area of ΔLMN? (7/8 + 1/8) = 9/8 in² **Practice B** **Developing Formulas for Circles and Regular Polygons** **Find each measurement. Give your answers in terms of π.** 1. A = πr² A = π(25)² A = 625π m² 2. A = πr² A = π(2a)² A = 4a²π in² 3. C = 2πr C = 2π(x + y) C = 2πx + 2πy yd 4. C = πd C = 1200π mi 5. the radius of OD in which C = 2π² cm C = 2πr 2π² = 2πr r = π 6. the diameter of OK in which A = (x² + 2x + 1)π km² A = πr² (x² + 2x + 1)π = πr² (x + 1)² = r² r = (x + 1) d = 2(x + 1) **Stella wants to cover a tabletop with nickels, dimes, or quarters. She decides to find which coin would cost the least to use.**   Stella measures the diameters of a nickel, a dime, and a quarter. They are 21.2 mm, 17.8 mm, and 24.5 mm. Find the areas of the nickel, the dime, and the quarter. Round to the nearest   tenth. Divide each coin’s value in cents by the coin’s area. Round to the nearest hundredth. Tell which coin has the least value per unit of area. Tell about how many nickels would cover a square tabletop that measures 1 square meter. Then find the cost of the coins. **Find the area of each regular polygon. Round to the nearest tenth.**   11. A = 1/2(ap) A = 1/2(10)(30√3) A = 150√3 ≈ 259.8 in² 12. A = 1/2(ap) A = 1/2(7)(4.9)(5) A ≈ 85.8 m²