Select the best answer for the question.
18. 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. 6 ft.
B. 36 ft.
C. [tex]$\sqrt{41 ft }$[/tex]
D. 12.81 ft.
Answer (2)
Using the Pythagorean theorem, we find that the length of the leg of the triangle is 6 feet. We calculate this by substituting the known values into the equation and solving for the unknown leg. Thus, the answer is A. 6 ft.
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Apply the Pythagorean theorem: a 2 + b 2 = c 2 , where a and b are the legs and c is the hypotenuse.
Substitute the given values: 8 2 + b 2 = 1 0 2 .
Solve for b : b = 1 0 2 − 8 2 = 36 .
Find the length of the other leg: 6 f t .
Explanation
Problem Analysis and Pythagorean Theorem 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. 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 (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.
Substitute Known Values Let a = 8 feet and c = 10 feet. We want to find the length of the other leg, b . Using the Pythagorean theorem, we have:
a 2 + b 2 = c 2
Substitute the given values:
8 2 + b 2 = 1 0 2
Solve 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
Find b To find b , we take the square root of both sides:
b = 36
b = 6 feet
Final Answer The length of the other leg is 6 feet. Therefore, the best answer is A. 6 ft.
Examples
The Pythagorean theorem is a fundamental concept in geometry and has many practical applications. For example, it can be used in construction to ensure that corners are square, in navigation to calculate distances, and in many other fields where right triangles are involved. Imagine you're building a ramp. If you know the height of the ramp (one leg) and the length of the base (another leg), you can use the Pythagorean theorem to calculate the length of the ramp itself (the hypotenuse). This ensures that the ramp is safe and meets the required specifications.
