To find the length of one leg in an isosceles right triangle with a given perimeter, let's follow these steps:
1. Define the Variables:
- Let [tex]\( a \)[/tex] be the length of one leg of the isosceles right triangle.
2. Properties of an Isosceles Right Triangle:
- In an isosceles right triangle, the two legs are equal in length.
- The hypotenuse can be expressed in terms of the legs by the Pythagorean theorem. If the leg length is [tex]\( a \)[/tex], then the hypotenuse [tex]\( h \)[/tex] is [tex]\( a\sqrt{2} \)[/tex].
3. Perimeter Expression:
- The perimeter of the triangle is the sum of the lengths of the two legs and the hypotenuse:
[tex]\[
\text{Perimeter} = a + a + a\sqrt{2} = 2a + a\sqrt{2}
\][/tex]
4. Given Perimeter:
- We are given that the perimeter is [tex]\( 94 + 94\sqrt{2} \)[/tex].
5. Set Up the Equation:
- Equate the perimeter expression to the given perimeter:
[tex]\[
2a + a\sqrt{2} = 94 + 94\sqrt{2}
\][/tex]
6. Solve for [tex]\( a \)[/tex]:
- Factor [tex]\( a \)[/tex] out from the left-hand side:
[tex]\[
a(2 + \sqrt{2}) = 94 + 94\sqrt{2}
\][/tex]
- Divide both sides of the equation by [tex]\( (2 + \sqrt{2}) \)[/tex]:
[tex]\[
a = \frac{94 + 94\sqrt{2}}{2 + \sqrt{2}}
\][/tex]
7. Simplify the Expression:
- Because [tex]\( \frac{94 + 94\sqrt{2}}{2 + \sqrt{2}} \)[/tex] is a bit complex due to the presence of the square root in the denominator, let's simplify the expression numerically.
- After simplifying, you'll find that:
[tex]\[
a \approx 66.468
\][/tex]
8. Conclusion:
- Thus, the length of one leg of the isosceles right triangle is approximately [tex]\( 66.468 \)[/tex] inches.
The correct answer is approximately [tex]\( 66.468 \)[/tex].
Hence, the answer is:
[tex]\[
\boxed{66.468}
\][/tex]
