The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures ?? $\sqrt{2}$ units. What is the length of one leg of the triangle?
11 units
$\11 \sqrt{2}$ units
22 units
$22 \sqrt{2}$ units
Answer (2)
In a 4 5 \textcirc 4 5 \textcirc 9 0 \textcirc triangle, the length of one leg is calculated using the relationship between the hypotenuse and the legs. Given the hypotenuse is 22 2 units, the length of one leg is determined to be 22 units. Thus, the length of one leg is 22 units.
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Recognize the relationship between the leg and hypotenuse in a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle: h = l 2 .
Substitute the given hypotenuse length: 22 2 = l 2 .
Divide both sides by 2 to solve for the leg length: l = 22 .
State the length of one leg: 22 units .
Explanation
Analyze the problem Let's analyze the problem. We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, which is a special type of right triangle where the two acute angles are both 4 5 ∘ . This means the two legs of the triangle are equal in length. We are given that the hypotenuse measures 22 2 units, and we need to find the length of one leg.
State the relationship between the leg and hypotenuse In a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, there's a special relationship between the length of each leg ( l ) and the length of the hypotenuse ( h ). The relationship is given by the formula: h = l 2 where:
h is the length of the hypotenuse
l is the length of one leg
Substitute the given value We are given that the hypotenuse h = 22 2 units. We can substitute this value into the formula: 22 2 = l 2 Now, we need to solve for l .
Solve for the leg length To find the length of one leg ( l ), we can divide both sides of the equation by 2 :
2 22 2 = 2 l 2 ⟹ l = 22 So, the length of one leg of the triangle is 22 units.
State the 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 put a diagonal fence across it to divide it into two equal right triangles. If the diagonal fence (the hypotenuse) needs to be 22 2 feet long, you can use the properties of a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle to determine that each side of your square garden should be 22 feet long. This ensures your garden is perfectly square and the diagonal fence divides it evenly.
