The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures $22 \sqrt{2}$ units. What is the length of one leg of the triangle?
Answer (1)
Recognize the triangle as a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, where both legs are equal.
Apply the Pythagorean theorem: x 2 + x 2 = ( 22 2 ) 2 .
Simplify and solve for x 2 : 2 x 2 = 968 ⟹ x 2 = 484 .
Find the length of one leg: x = 484 = 22 .
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle with a hypotenuse of length 22 2 units. We need to find the length of one of the legs of the triangle.
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 lengths of the two legs is equal to the square of the length of the hypotenuse. Therefore, we have:
x 2 + x 2 = ( 22 2 ) 2
Simplifying the Equation Now, we simplify the equation:
2 x 2 = ( 22 2 ) 2
2 x 2 = 2 2 2 × ( 2 ) 2
2 x 2 = 484 × 2
2 x 2 = 968
Solving for x^2 Next, we solve for x 2 :
x 2 = 2 968
x 2 = 484
Solving for x Finally, we solve for x by taking the square root of both sides:
x = 484
x = 22
Therefore, the length of one leg of the triangle is 22 units.
Final Answer The length of one leg of the 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle is 22 units.
Examples
Imagine you're building a square-shaped garden and want to put a diagonal fence across it to divide it into two equal right triangles. If the length of the diagonal fence (the hypotenuse) is 22 2 feet, you can use the properties of a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle to determine that each side of the garden (the legs of the triangle) should be 22 feet long. This ensures your garden is perfectly square and equally divided.
