Select the best answer for the question.
6. 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. 12.81 ft
B. [tex]$\sqrt{41 ft }$[/tex]
C. 6 ft
D. 36 ft
Answer (1)
We have a right triangle with one leg of 8 feet and a hypotenuse of 10 feet.
Using the Pythagorean theorem, a 2 + b 2 = c 2 , where a and b are the legs and c is the hypotenuse.
Substituting the given values, we get a 2 + 8 2 = 1 0 2 , which simplifies to a 2 + 64 = 100 .
Solving for a , we find a = 36 = 6 feet. Therefore, the length of the other leg is 6 f t .
Explanation
Problem Analysis 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.
Applying the Pythagorean Theorem Let a and b be the lengths of the legs of the right triangle, and c be the length of the hypotenuse. The Pythagorean theorem states that a 2 + b 2 = c 2 . We are given that b = 8 feet and c = 10 feet. We need to find a .
Substitution Substitute the given values into the Pythagorean theorem:
a 2 + 8 2 = 1 0 2
a 2 + 64 = 100
Isolating a^2 Solve for a 2 :
a 2 = 100 − 64
a 2 = 36
Finding a Take the square root of both sides to find a :
$a =
\sqrt{36}$
a = 6 feet
Final Answer The length of the other leg is 6 feet. Therefore, the best answer is C. 6 ft.
Examples
The Pythagorean theorem is a fundamental concept in construction and navigation. For example, builders use it to ensure that corners of buildings are square, and navigators use it to calculate distances and courses. Imagine you're building a ramp. If you know the height (one leg) and the length of the base (another leg), you can use the Pythagorean theorem to find the length of the ramp (the hypotenuse). This ensures your ramp is safe and meets the required specifications.
