To find the length of the hypotenuse of an isosceles right triangle, we can use the Pythagorean theorem, which is stated as a^2 + b^2 = c^2 for a triangle with sides of lengths a, b, and c, where c is the length of the hypotenuse.
For an isosceles right triangle, the lengths of the two legs (sides a and b) are equal. Since we are given that each leg of the triangle has a length of \(15\sqrt{2}\) ft, we can let a = b = \(15\sqrt{2}\) ft.
Now, we substitute these values into the Pythagorean theorem to find the length of the hypotenuse (c):
\((15\sqrt{2})^2 + (15\sqrt{2})^2 = c^2\)
Squaring \(15\sqrt{2}\) gives us:
\((15\sqrt{2})^2 = 15^2 \times (\sqrt{2})^2 = 225 \times 2 = 450\)
Therefore, we have:
\(450 + 450 = c^2\)
Adding these together gives us:
\(900 = c^2\)
We take the square root of both sides of the equation to find the value of c:
\(\sqrt{900} = c\)
The square root of 900 is 30, so:
\(c = 30\)
Hence, the length of the hypotenuse of an isosceles right triangle with legs of \(15\sqrt{2}\) ft each is 30 feet.
The correct answer from the multiple choices given is 30 feet.
