The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures 4 cm. What is the length of one leg of the triangle?
A. 2 cm
B. $2 \sqrt{2} cm$
C. 4 cm
D. $4 \sqrt{2} cm$
Answer (1)
Recognize that in a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, both legs are equal.
Apply the Pythagorean theorem: x 2 + x 2 = 4 2 .
Simplify and solve for x : x = 8 = 2 2 .
State the length of one leg: 2 2 cm .
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle with a hypotenuse of 4 cm. Our goal is to find the length of one of the legs.
Applying the Pythagorean Theorem 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 . According to the Pythagorean theorem, the sum of the squares of the legs is equal to the square of the hypotenuse. Therefore, we have:
x 2 + x 2 = 4 2
Simplifying the Equation Simplifying the equation, we get:
2 x 2 = 16
Dividing both sides by 2:
x 2 = 8
Solving for x Taking the square root of both sides:
x = 8
Since 8 = 4 × 2 , we can simplify the square root as follows:
x = 4 × 2 = 4 × 2 = 2 2
Final Answer Therefore, the length of one leg of the triangle is 2 2 cm.
Examples
Right triangles are fundamental in construction and engineering. For example, when building a ramp, knowing the hypotenuse (the length of the ramp) and the angle allows you to calculate the necessary height and base length, ensuring the ramp meets safety standards and is easy to use. In this case, a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle helps determine equal height and base, simplifying design and construction.
