University:
High School
Course:
High School Mathematics
Academic year:

2022
Views:

494

Pages:

2
Author:

kortmouebf7y

Theorems Theorem 9.4 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg. Examples: 2² + 2² = x² 3² + 3² = x² 7² + 7² = x² 2√2 = x 3√2 = x 7√2 = x Diagram: A 45°-45°-90° triangle with sides labeled w, w, and w√2. Theorem 9.5 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. Example: 5² + x² = 10² 25 + x² = 100 x = 5√3 Diagram: A 30°-60°-90° triangle with sides labeled w, w√3, and 2w. Extra Practice In Exercises 1-4, find the value of x. Write your answer in simplest form. Diagram: A 45°-45°-90° triangle with sides labeled 10, 10, and x. Solution: x = 10√2 Diagram: A 45°-45°-90° triangle with sides labeled 1/2, 1/2, and x. Solution: x = √2/2 Diagram: A 30°-60°-90° triangle with sides labeled 6, 6√3, and x. Solution: x = 12 Diagram: A 30°-60°-90° triangle with sides labeled 12, x, and 6√2. Solution: x = 6√3 In Exercises 5-7, find the values of x and y. Write your answers in simplest form. 5. * Diagram: A 30-60-90 triangle with sides labeled 15, x, and y. * Solution: * y = 2(15) * y = 30 * x = 15√3 6. * Diagram: A 45-45-90 triangle with sides labeled 8, 8, and y. * Solution: * y = 8√2 * x = 8 7. * Diagram: A 30-60-90 triangle with sides labeled 2.2, x, and y. * Solution: * y = 2(2.2) * y = 4.4 * x = 2.2√3 In Exercises 8 and 9, sketch the figure that is described. Find the indicated length. 8. * Description: The length of a diagonal in a square is 32 inches. Find the perimeter of the square. * Diagram: A square with a diagonal of 32 inches. * Solution: * Perimeter = 4(side) * Side = 32/√2 = 16√2 * Perimeter = 4(16√2) = 64√2 inches 9. * Description: An isosceles triangle with 30° base angles has an altitude of √3 meters. Find the length of the base of the isosceles triangle. * Diagram: An isosceles triangle with a 30-60-90 triangle formed by the altitude. * Solution: * Base = 2(√3) = 2√3 meters 10. Find the area of ΔDEF. Round decimal answers to the nearest tenth. * Diagram: A triangle with sides labeled 14 m, 7√3 m, and 7√3 m. * Solution: * Area = (1/2)(base)(height) * Area = (1/2)(14)(7√3) * Area = 49√3 ≈ 84.9 m²