To find the length of a leg in a right triangle when one leg and the hypotenuse are known, we use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse ([tex]\( c \)[/tex]) is equal to the sum of the squares of the lengths of the other two legs ([tex]\( a \)[/tex] and [tex]\( b \)[/tex]). Mathematically, this is represented as:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
In this problem, we are given:
- One leg ([tex]\( b \)[/tex]) is 8 feet long.
- The hypotenuse ([tex]\( c \)[/tex]) is 10 feet long.
We need to find the length of the other leg ([tex]\( a \)[/tex]). According to the Pythagorean theorem:
[tex]\[ a^2 + 8^2 = 10^2 \][/tex]
First, calculate the squares of the given numbers:
[tex]\[ 8^2 = 64 \][/tex]
[tex]\[ 10^2 = 100 \][/tex]
Substitute these values into the equation:
[tex]\[ a^2 + 64 = 100 \][/tex]
Next, isolate [tex]\( a^2 \)[/tex] by subtracting 64 from both sides of the equation:
[tex]\[ a^2 = 100 - 64 \][/tex]
[tex]\[ a^2 = 36 \][/tex]
To find [tex]\( a \)[/tex], take the square root of both sides:
[tex]\[ a = \sqrt{36} \][/tex]
[tex]\[ a = 6 \][/tex]
Therefore, the length of the other leg is 6 feet.
The best answer is:
B. 6 ft
