The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures 18 cm. What is the length of one leg of the triangle?
A. 9 cm
B. $9 \sqrt{2} cm$
C. 18 cm
D. $18 \sqrt{2} cm$
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 = 1 8 2 .
Solve for x : x = 162 .
Simplify the result: x = 9 2 .
The length of one leg of the triangle is 9 2 c m .
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle with a hypotenuse of 18 cm. We need to find the length of one of the legs. Since it's a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, the two legs are of equal length. Let's call the length of each leg 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. Therefore, we have:
x 2 + x 2 = 1 8 2
Simplifying the Equation Simplifying the equation, we get:
2 x 2 = 324
Isolating x 2 Dividing both sides by 2, we have:
x 2 = 162
Solving for x Taking the square root of both sides, we get:
x = 162
Simplifying the Square Root We can simplify 162 as follows:
162 = 81 ⋅ 2 = 81 ⋅ 2 = 9 2
Final Answer Therefore, the length of one leg of the triangle is 9 2 cm.
Examples
Understanding 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangles is very useful in construction and design. For example, if you are building a square structure and need to brace it diagonally, the diagonal brace forms a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle. If you know the length of the brace (the hypotenuse), you can easily calculate the required side lengths to ensure the structure is perfectly square. This principle helps ensure accuracy and stability in various construction projects.
