The hypotenuse of a 45-45-90 triangle has a length of 10 units. What is the length of one of its legs?
A. [tex]$10 \sqrt{2}$[/tex] units
B. 10 units
C. [tex]$5 \sqrt{2}$[/tex] units
D. 5 units
Answer (2)
Recognize the properties of a 45-45-90 triangle.
Apply the Pythagorean theorem: x 2 + x 2 = 1 0 2 .
Solve for the leg length x : x = 50 = 5 2 .
The length of one leg is 5 2 units.
Explanation
Problem Analysis We are given a 45-45-90 triangle with a hypotenuse of length 10 units. Our goal is to find the length of one of the legs of this triangle.
Applying the Pythagorean Theorem In a 45-45-90 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 legs is equal to the square of the hypotenuse. Therefore, we have: x 2 + x 2 = 1 0 2
Simplifying the Equation Simplifying the equation, we get: 2 x 2 = 100
Isolating x 2 Dividing both sides by 2, we have: x 2 = 50
Solving for x Taking the square root of both sides, we get: x = 50 We can simplify 50 as follows: x = 25 × 2 = 25 × 2 = 5 2
Final Answer Therefore, the length of one leg of the 45-45-90 triangle is 5 2 units.
Examples
45-45-90 triangles are commonly used in construction and design. For example, if you are building a ramp that needs to rise at a 45-degree angle, and you want the horizontal distance covered by the ramp to be 5 feet, then the height of the ramp will also be 5 feet, and the length of the ramp (the hypotenuse) will be 5 2 feet. This understanding helps in precise measurements and structural calculations.
The length of one leg of the 45-45-90 triangle, given a hypotenuse of 10 units, is 5 2 units. The answer is option C. This is derived using the properties of a 45-45-90 triangle and the Pythagorean theorem.
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