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 the triangle as a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle.
Apply the Pythagorean theorem: x 2 + x 2 = 4 2 .
Solve for x : x = 8 = 2 2 .
State the length of one leg: 2 2 c m .
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle with a hypotenuse of 4 cm. We need 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
Isolating x 2 Dividing both sides by 2, we have: x 2 = 8
Solving for x Taking the square root of both sides, we get: x = 8 x = 4 × 2 x = 2 2
Final Answer Therefore, the length of one leg of the triangle is 2 2 cm.
Examples
Understanding 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangles is useful in construction and design. For example, if you're building a square structure and need to brace it diagonally, the diagonal brace forms a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle. Knowing the length of the sides allows you to calculate the necessary length of the brace. In this case, if the sides of the square are 2 2 cm each, the diagonal brace would need to be 4 cm long.
