In an isosceles right triangle, the lengths of the two legs are equal, and the relationship between the legs and the hypotenuse can be derived from applying the Pythagorean theorem.
Given:
- Each leg of the isosceles right triangle has a length of [tex]\( 15 \sqrt{2} \)[/tex] feet.
- Let [tex]\( L \)[/tex] denote the length of each leg.
Since it's an isosceles right triangle, we can use the Pythagorean theorem to find the length of the hypotenuse [tex]\( H \)[/tex]:
[tex]\[
H = \sqrt{L^2 + L^2}
\][/tex]
First, substitute [tex]\( L = 15 \sqrt{2} \)[/tex]:
[tex]\[
H = \sqrt{(15 \sqrt{2})^2 + (15 \sqrt{2})^2}
\][/tex]
Calculate the square of [tex]\( 15 \sqrt{2} \)[/tex]:
[tex]\[
(15 \sqrt{2})^2 = 15^2 \cdot (\sqrt{2})^2 = 225 \cdot 2 = 450
\][/tex]
Now, sum the squares of the legs (since they are equal and there are two of them):
[tex]\[
H = \sqrt{450 + 450} = \sqrt{900}
\][/tex]
Simplify the square root:
[tex]\[
H = \sqrt{900} = 30
\][/tex]
Therefore, the length of the hypotenuse is:
[tex]\[
\boxed{30 \text{ feet}}
\][/tex]
