The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures $7 \sqrt{2}$ units. What is the length of one leg of the triangle?
A. 7 units
B. $7 \sqrt{2}$ units
C. 14 units
D. $14 \sqrt{2}$ units
Answer (1)
Recognize the properties of a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle.
Apply the Pythagorean theorem: x 2 + x 2 = ( 7 2 ) 2 .
Simplify and solve for x : 2 x 2 = 98 ⟹ x 2 = 49 ⟹ x = 7 .
Conclude that the length of one leg is 7 units.
Explanation
Problem Analysis We are given a 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle with a hypotenuse of length 7 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 square of the hypotenuse is equal to the sum of the squares of the other two sides (legs). Therefore, we have:
x 2 + x 2 = ( 7 2 ) 2
Simplifying the Equation Simplifying the equation, we get:
2 x 2 = ( 7 2 ) 2
2 x 2 = 49 × 2
2 x 2 = 98
Solving for x 2 Now, we solve for x 2 :
x 2 = 2 98
x 2 = 49
Finding the Length of the Leg Taking the square root of both sides to find x :
x = 49
x = 7
Final Answer Therefore, the length of one leg of the 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangle is 7 units.
Examples
Understanding 4 5 ∘ − 4 5 ∘ − 9 0 ∘ triangles is useful in construction and design. For example, if you're building a ramp that needs to rise at a 4 5 ∘ angle, and you know the horizontal distance the ramp covers is 7 feet, then you also know the vertical rise will be 7 feet, and the length of the ramp itself will be 7 2 feet. This ensures your ramp meets the required specifications.
