The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures 128 cm. What is the length of one leg of the triangle?
A. 64 cm
B. $64 \sqrt{2} cm$
C. 128 cm
D. $128 \sqrt{2} cm$
Answer (1)
Recognize the triangle as a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, meaning its legs are equal.
Apply the Pythagorean theorem: x 2 + x 2 = 12 8 2 .
Solve for x : x = 2 12 8 2 = 64 2 .
The length of one leg is 64 2 c m .
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle with a hypotenuse of 128 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 one leg x .
Applying the 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 = 12 8 2
Simplifying the Equation Simplifying the equation, we get: 2 x 2 = 12 8 2
Solving for x^2 Now, we solve for x 2 :
x 2 = 2 12 8 2 x 2 = 2 16384 x 2 = 8192
Solving for x Taking the square root of both sides to solve for x :
x = 8192 x = 2 ∗ 4096 x = 2 ∗ 6 4 2 x = 64 2
Final Answer Therefore, the length of one leg of the triangle is 64 2 cm.
Examples
Imagine you're building a square frame and need to add a diagonal support to make it stronger. If the sides of the square are equal to the legs of a 45-45-90 triangle, the diagonal support acts as the hypotenuse. Knowing the length of the diagonal (hypotenuse) helps you determine the required length of the sides (legs) to ensure the frame is perfectly square and stable. This principle is used in construction, carpentry, and even in designing furniture to ensure structural integrity.
