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Consider the planned triangle (T

_{0}) to have sides 8a, 8a, 8b. The semi-perimeter is 8a + 4b, and hence the area A, by Heron’s formula, is:The actually constructed triangle (T

_{1}) has sides 8a, 8a, 6b. The semi-perimeter 8a + 3b, and the area is also A. Heron’s formula gives:Hence:

which simplifies to:

So the semi-perimeter of T

_{0}is 9b, and the semi-perimeter of T_{1}is 8b.Now, considering the incircles of the triangles. The area of a triangle is the product of the inradius and the semi-perimeter, so:

We see that the triangles are arranged as below:

And so the area is:

And so the sides of T

_{0}are (30, 30, 48) and the sides of T_{1}are (30, 30, 36), which is the required answer.Solution:The lengths of fencing bought were two lengths of 30 feet and one length of 36 feet.