Using the Pythagorean theorem, find the length of a leg of a right triangle if the other leg is 8 feet long and the hypotenuse is 10 feet long.
A) $\sqrt{41 ft }$.
B) 36 ft .
C) 12.81 ft .
D) 6 ft .
Answer (1)
We have a right triangle with one leg of 8 feet and a hypotenuse of 10 feet.
Apply the Pythagorean theorem: a 2 + b 2 = c 2 , where a = 8 and c = 10 .
Solve for the unknown leg b : b = c 2 − a 2 = 1 0 2 − 8 2 = 36 .
The length of the other leg is 6 feet.
Explanation
Problem Analysis and Setup We are given a right triangle with one leg of length 8 feet and a hypotenuse of length 10 feet. We need to find the length of the other leg using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). This can be written as a 2 + b 2 = c 2 , where a and b are the lengths of the legs, and c is the length of the hypotenuse.
Applying the Pythagorean Theorem Let a = 8 feet and c = 10 feet. We want to find the length of the other leg, b . We can plug the given values into the Pythagorean theorem:
8 2 + b 2 = 1 0 2
Solving for b^2 Now, we solve for b 2 :
64 + b 2 = 100
Subtract 64 from both sides:
b 2 = 100 − 64
b 2 = 36
Finding b To find b , we take the square root of both sides:
$b =
36
b = 6 feet
Final Answer Therefore, the length of the other leg is 6 feet.
Examples
The Pythagorean theorem is a fundamental concept in geometry and has many real-world applications. For example, architects and engineers use it to ensure that buildings are square and stable. Imagine you're building a ramp. If you know the height and the horizontal distance the ramp needs to cover, you can use the Pythagorean theorem to calculate the length of the ramp itself, ensuring it meets safety standards.
