Each leg of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures 12 cm. What is the length of the hypotenuse?
A. 6 cm
B. $6 \sqrt{2} cm$
C. 12 cm
D. $12 \sqrt{2} cm$
Answer (1)
Recognize the triangle as a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle.
Recall the relationship between the leg length ( l ) and the hypotenuse length ( h ): h = l 2 .
Substitute the given leg length, l = 12 cm, into the formula.
Calculate the hypotenuse length: 12 2 cm .
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, where each leg measures 12 cm. Our goal is to find the length of the hypotenuse.
Formula for Hypotenuse In a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, there's a special relationship between the legs and the hypotenuse. If the length of each leg is l , then the length of the hypotenuse h is given by the formula:
h = l 2
In our case, the length of each leg is l = 12 cm.
Calculate Hypotenuse Now, we substitute the given value of l into the formula:
h = 12 2
Therefore, the length of the hypotenuse is 12 2 cm.
Final Answer The length of the hypotenuse of the 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle is 12 2 cm.
Examples
Imagine you're building a square frame and need to add a diagonal brace for extra support. If each side of the square is 12 cm, the diagonal brace (which forms the hypotenuse of a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle) needs to be 12 2 cm long. This ensures the frame is sturdy and maintains its shape.
