The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures $10 \sqrt{5} in$. What is the length of one leg of the triangle?
A. $5 \sqrt{10}$
B. $10 \sqrt{5}$
C. $5 \sqrt{5}$
D. $10 \sqrt{10}$
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 = ( 10 5 ) 2 .
Simplify and solve for x 2 : 2 x 2 = 500 , so x 2 = 250 .
Find the length of one leg by taking the square root: x = 250 = 5 10 .
The length of one leg of the triangle is 5 10 .
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle with a hypotenuse of 10 5 inches. Our goal is to find the length of one of the legs of the triangle. 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, we know that the sum of the squares of the legs is equal to the square of the hypotenuse. In this case, we have:
x 2 + x 2 = ( 10 5 ) 2
Simplifying the Equation Now, let's simplify the equation:
2 x 2 = ( 10 5 ) 2
2 x 2 = 1 0 2 × ( 5 ) 2
2 x 2 = 100 × 5
2 x 2 = 500
Solving for x^2 Next, we solve for x 2 :
x 2 = 2 500
x 2 = 250
Finding x Now, we take the square root of both sides to find x :
x = 250
x = 25 × 10
x = 25 × 10
x = 5 10
Final Answer Therefore, the length of one leg of the triangle is 5 10 inches.
Examples
Imagine you're building a ramp for skateboarding, and you want the ramp to form a 4 5 ∘ angle with the ground. If you know the length of the ramp (the hypotenuse), you can use the properties of a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle to calculate the height and the horizontal distance the ramp covers. This ensures your ramp is safe and has the desired incline.
