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. $5 \sqrt{5}$
B. $5 \sqrt{10}$
C. $10 \sqrt{5}$
D. $10 \sqrt{10}$
Answer (1)
Recognize the properties of a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, where the two legs are equal and the hypotenuse is 2 times the length of a leg.
Set up the equation x 2 = 10 5 , where x is the length of one leg.
Solve for x by dividing both sides by 2 and rationalizing the denominator.
Find the length of one leg: 5 10 .
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle with a hypotenuse of length 10 5 inches. We need to find the length of one of the legs of the triangle.
Setting up the relationship In 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 . The hypotenuse is then given by x 2 .
Solving for the leg length We are given that the hypotenuse has a length of 10 5 inches. Therefore, we can set up the equation: x 2 = 10 5 To solve for x , we divide both sides of the equation by 2 :
x = 2 10 5 To rationalize the denominator, we multiply both the numerator and the denominator by 2 :
x = 2 ⋅ 2 10 5 ⋅ 2 = 2 10 10 = 5 10 So, the length of one leg of the triangle is 5 10 inches.
Final Answer Therefore, the length of one leg of the 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle is 5 10 inches.
Examples
Understanding 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangles is very useful in construction and design. For example, if you are building a ramp that needs to have a 4 5 ∘ angle, and you know the height of the ramp needs to be a certain value, you can easily calculate the length of the base and the hypotenuse using the properties of these special triangles. In this case, if you knew the hypotenuse was 10 5 inches, you could quickly determine that each leg (height and base) should be 5 10 inches.
