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 (2)
The length of one leg of a 4 5 ° − 4 5 ° − 9 0 ° triangle with a 4 cm hypotenuse is determined to be 2 2 cm by using the relationship between the legs and hypotenuse. Therefore, the chosen answer is B. 2 2 cm.
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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 = 4 2 .
Simplify and solve for x 2 : 2 x 2 = 16 ⟹ x 2 = 8 .
Find the length of one leg by taking the square root: x = 8 = 2 2 cm. The length of one leg of the triangle is 2 2 c m .
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle with a hypotenuse of 4 cm. Our goal is 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 In a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle, the relationship between the legs and the hypotenuse can be expressed using the Pythagorean theorem: a 2 + b 2 = c 2 , where a and b are the lengths of the legs, and c is the length of the hypotenuse. In our case, a = x , b = x , and c = 4 . So, we have:
x 2 + x 2 = 4 2
Simplifying the Equation Now, let's simplify the equation:
2 x 2 = 16
Solving for x 2 Next, we solve for x 2 :
x 2 = 2 16
x 2 = 8
Solving for x Now, we solve for x by taking the square root of both sides:
x = 8
x = 4 × 2
x = 2 2 cm
Final Answer Therefore, the length of one leg of the triangle is 2 2 cm.
Examples
Imagine you're building a square-shaped garden and want to divide it diagonally with a pathway. If the diagonal pathway is 4 meters long, this problem helps you calculate the length of each side of the square garden. Knowing the relationship between the diagonal (hypotenuse) and sides (legs) of a 45-45-90 triangle allows you to determine the garden's dimensions easily, ensuring your design fits perfectly within your available space.
