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?
A. 11 units
B. $11 \sqrt{2}$ units
C. 22 units
D. $22 \sqrt{2}$ units
Answer (2)
The length of one leg of the 4 5 circ − 4 5 circ − 9 0 circ triangle with a hypotenuse of 22 2 units is 22 units.
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Use the Pythagorean theorem for a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle: x 2 + x 2 = ( 22 2 ) 2 .
Simplify the equation: 2 x 2 = 968 .
Solve for x 2 : 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 22 2 units. Our goal is to find the length of one of the legs of the triangle.
Applying the Pythagorean Theorem Let's denote the length of one leg of the triangle as x . Since it's a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, both legs have the same length. We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In our case, this can be written as:
x 2 + x 2 = ( 22 2 ) 2
Simplifying the Equation Now, let's 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
Finding the Length of the Leg Finally, we take the square root of both sides to find x :
x = 484
x = 22
Final Answer Therefore, 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 divide it diagonally with a path. If the diagonal path measures 22 2 feet, this problem helps you determine the length of each side of the square garden. Knowing the side length is crucial for planning the layout, purchasing materials, and ensuring the garden fits perfectly in your yard. In this case, each side of the garden would be 22 feet long.
