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) 6 ft .
C) 12.81 ft .
D) 36 ft .
Answer (2)
The problem requires finding the length of a leg in a right triangle using the Pythagorean theorem.
The Pythagorean theorem is a 2 + b 2 = c 2 , where a and b are the legs and c is the hypotenuse.
Substitute the given values: a 2 + 8 2 = 1 0 2 , which simplifies to a 2 = 36 .
Solve for a: a = \[ \sqrt{36} \] = 6 . 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. Let's call the unknown leg 'a'.
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.
Substitution In our case, we know that one leg (b) is 8 feet long and the hypotenuse (c) is 10 feet long. We can substitute these values into the Pythagorean theorem: a 2 + 8 2 = 1 0 2 .
Solving for a^2 Now we can solve for a 2 : a 2 + 64 = 100 . Subtracting 64 from both sides, we get a 2 = 100 − 64 = 36 .
Finding a To find the length of leg 'a', we take the square root of both sides of the equation: a = \[ \sqrt{36} \] = 6 .
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, it can be used in construction to ensure that buildings are square and stable. Imagine you're building a rectangular shed. You can use the Pythagorean theorem to check if the corners are right angles. If the sides are 3 meters and 4 meters long, the diagonal should be 5 meters long (since 3 2 + 4 2 = 5 2 ). This ensures the shed is perfectly rectangular.
The length of the unknown leg in the right triangle is 6 feet, found using the Pythagorean theorem. This is calculated by substituting the known values into the equation and solving for the unknown leg. The correct answer is B) 6 ft.
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