Mark wants to make a unique triangular design that will cover the vacant space of his bedroom wall. His mother, Annie, Suggests that hypotenuse should be at least 15ft and one leg is 3ft shorter than the order. What is the length of the long leg?
Answer (1)
Pythagorean Theorem1. Define the variablesLet a be the length of the shorter leg.Let b be the length of the longer leg.Let c be the length of the hypotenuse (given as 15 ft).2. Write the given information as equationsc = 15 ft (hypotenuse)a = b - 3 ft (the shorter leg is 3 ft shorter than the longer leg)3. Apply the Pythagorean theorema² + b² = c²4. Substitute the known values and the expression for a into the Pythagorean theorem(b - 3)² + b² = 15²5. Expand and simplify the equation(b² - 6b + 9) + b² = 2252b² - 6b + 9 = 2252b² - 6b - 216 = 0b² - 3b - 108 = 0(b - 1²)(b + 9) = 0b - 1² = 0 or b + 9 = 0b = 1² or b = -96. Choose the valid solutionSince the length of a leg cannot be negative, we discard b = -9.Therefore, b = 1² ft.Final answer:The length of the longer leg is 1 ft.
