The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures 18 cm. What is the length of one leg of the triangle?
A. 9 cm
B. $9 \sqrt{2} cm$
C. 18 cm
D. $18 \sqrt{2} cm$
Answer (1)
Recognize the triangle as a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle and denote the length of each leg as x .
Apply the Pythagorean theorem: x 2 + x 2 = 1 8 2 .
Solve for x : x = 162 = 9 2 .
The length of one leg of the triangle is 9 2 cm .
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle with a hypotenuse of 18 cm. We need to find the length of one of the legs. Since it's a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, the two legs are of equal length. Let's call the length of each leg x .
Apply Pythagorean Theorem Using the Pythagorean theorem, we know that the sum of the squares of the legs is equal to the square of the hypotenuse. Therefore, we have: x 2 + x 2 = 1 8 2
Simplify the Equation Simplifying the equation, we get: 2 x 2 = 324
Isolate x 2 Dividing both sides by 2, we have: x 2 = 162
Solve for x Taking the square root of both sides, we get: x = 162 We can simplify the square root by factoring out the largest perfect square: x = 81 ⋅ 2 x = 81 ⋅ 2 x = 9 2
Final Answer Therefore, the length of one leg of the triangle is 9 2 cm.
Examples
Imagine you're building a square-shaped garden and want to divide it diagonally with a pathway. If the diagonal pathway measures 18 meters, this problem helps you calculate the length of each side of the square garden. Knowing the side length is crucial for planning the layout, purchasing materials, and ensuring the garden fits perfectly in your yard. The length of each side would be 9 2 meters, which is approximately 12.73 meters.
