To solve the problem of determining the length of the other leg of a right triangle when one leg and the hypotenuse are given, we will use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Mathematically, this is expressed as:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse.
Given:
- One leg \( a = \sqrt{7} \)
- Hypotenuse \( c = 4 \)
We need to find the length of the other leg \( b \).
1. First, we square the lengths of the leg and the hypotenuse:
[tex]\[ a^2 = (\sqrt{7})^2 = 7 \][/tex]
[tex]\[ c^2 = 4^2 = 16 \][/tex]
2. Next, we use the Pythagorean theorem to set up the equation and solve for \( b^2 \):
[tex]\[ a^2 + b^2 = c^2 \][/tex]
[tex]\[ 7 + b^2 = 16 \][/tex]
3. Subtract \( 7 \) from both sides to solve for \( b^2 \):
[tex]\[ b^2 = 16 - 7 \][/tex]
[tex]\[ b^2 = 9 \][/tex]
4. Finally, take the square root of both sides to find \( b \):
[tex]\[ b = \sqrt{9} = 3 \][/tex]
Therefore, the length of the other leg of the right triangle is \( 3 \). The correct answer is:
[tex]\[ 3 \][/tex]
