Final answer:
The length of one leg of the isosceles right triangle is 94 - 47√2 inches.
Step-by-step explanation:
To find the length of one leg of an isosceles right triangle with a given perimeter, we first need to find the length of the hypotenuse. The hypotenuse can be found using the formula for the perimeter of an isosceles right triangle: P = 2a + c, where P is the perimeter, a is the length of one leg, and c is the length of the hypotenuse. In this case, the perimeter is given as 94+94√2 inches, so we can set up the equation as 94+94√2 = 2a + c. Since the triangle is isosceles, we know that the lengths of the two legs are equal, so we can denote the length of one leg as a. The formula for the hypotenuse in a right triangle is c = √(a² + a²), which simplifies to c = a√2. Substituting c = a√2 into the first equation, we have 94+94√2 = 2a + a√2. To solve for a, we can isolate the term with a by subtracting a√2 from both sides of the equation, resulting in 94 = (2 + √2)a. Dividing both sides of the equation by (2 + √2), we have a = 94 / (2 + √2). To simplify this expression, we can rationalize the denominator by multiplying both the numerator and denominator by (2 - √2), resulting in a = 94(2 - √2) / (2 + √2)(2 - √2). Simplifying this further, we have a = 188 - 94√2 / (4 - 2). Finally, we can simplify this expression to a = 94 - 47√2 inches.