To determine the length of the missing leg of a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that, in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Mathematically, it is expressed as:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the lengths of the legs, and [tex]\(c\)[/tex] is the length of the hypotenuse.
Given:
- One leg [tex]\( a = \sqrt{6} \)[/tex]
- Hypotenuse [tex]\( c = 7 \)[/tex]
We need to find the length of the other leg [tex]\( b \)[/tex].
1. First, we square the given lengths:
[tex]\[ a^2 = (\sqrt{6})^2 = 6 \][/tex]
[tex]\[ c^2 = 7^2 = 49 \][/tex]
2. Substitute these values into the Pythagorean theorem formula and solve for [tex]\( b^2 \)[/tex]:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
[tex]\[ 6 + b^2 = 49 \][/tex]
3. Solve for [tex]\( b^2 \)[/tex]:
[tex]\[ b^2 = 49 - 6 \][/tex]
[tex]\[ b^2 = 43 \][/tex]
4. Finally, take the square root of both sides to solve for [tex]\( b \)[/tex]:
[tex]\[ b = \sqrt{43} \][/tex]
Therefore, the length of the other leg is [tex]\( \sqrt{43} \)[/tex].
