In an isosceles right triangle, the two legs are equal in length. Let's assume that the length of each leg of the original triangle is "x" inches. Since it's an isosceles right triangle, the hypotenuse is also "x" inches.
Given that the third side of the original triangle measures 25 inches, we can set up the following equation based on the perimeter:
x + x + 25 = 85
Simplifying the equation:
2x + 25 = 85
Subtracting 25 from both sides:
2x = 60
Dividing both sides by 2:
x = 30
So, the length of each leg of the original triangle is 30 inches.
Now, let's consider the dilated triangle. We are told that the leg of the dilated triangle measures 6 inches. Let's assume the scale factor of the dilation is "k."
The length of each leg of the dilated triangle can be found by multiplying the length of the corresponding leg of the original triangle by the scale factor:
Leg of dilated triangle = k * Leg of original triangle
Leg of dilated triangle = k * 30
Given that the leg of the dilated triangle measures 6 inches, we can set up the following equation:
k * 30 = 6
Dividing both sides by 30:
k = 6/30
Simplifying:
k = 1/5
So, the scale factor of the dilation is 1/5.
Now, to find the perimeter of the dilated triangle, we need to multiply the length of each side of the original triangle by the scale factor and sum them up:
Perimeter of dilated triangle = 2 * (k * 30) + k * 25
Plugging in the values:
Perimeter of dilated triangle = 2 * (1/5 * 30) + 1/5 * 25
Simplifying:
Perimeter of dilated triangle = 2 * 6 + 5
Perimeter of dilated triangle = 12 + 5
Perimeter of dilated triangle = 17 inches
Therefore, the perimeter of the dilated triangle is 17 inches.