The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures 24 inches. What is the length of one of the legs of the triangle?
A. 12 in.
B. $12 \sqrt{2}$ in.
C. 24 in.
D. $24 \sqrt{2}$ in.
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 = 2 4 2 .
Solve for x : x = 2 2 4 2 = 12 2 .
The length of one leg is 12 2 in.
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle with a hypotenuse of 24 inches. Our goal is 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 denote the length of each leg as x .
Applying the Pythagorean Theorem Using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs). In our case, the theorem can be written as:
x 2 + x 2 = 2 4 2
Simplifying the Equation Now, let's simplify the equation:
2 x 2 = 2 4 2
2 x 2 = 576
Isolating x 2 To solve for x 2 , divide both sides of the equation by 2:
x 2 = 2 576
x 2 = 288
Solving for x Now, take the square root of both sides to find x :
x = 288
We can simplify 288 by factoring out the largest perfect square. Since 288 = 144 × 2 , we have:
x = 144 × 2
x = 144 × 2
x = 12 2
Final Answer Therefore, the length of one of the legs of the 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle is 12 2 inches.
Examples
Imagine you're building a square-shaped garden and want to divide it diagonally with a pathway. If the diagonal path measures 24 feet, this problem helps you calculate the length of each side of the square garden, ensuring you have the right dimensions for your project. This is a practical application of understanding 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangles in everyday construction and design.
