The product of the lengths of the sides of an isosceles triangle is 64cm, and the perimeter is 12cm. How long is one side?

Let’s denote by x the side of an isosceles triangle. For y we denote the base of the triangle.
Let’s compose a system of equations:

x * x * y = 64;

2x + y = 12;

Let us express the variable y in terms of x:

x * x * y = 64;

y = 12 – 2x;

Substitute the resulting variable y from the second expression into the first equation:

x * x * (12 – 2x) = 64;

Let’s open the brackets and solve the resulting equation using the factorization of the polynomial. We will get 2 roots, but one of them cannot be the answer, since it is negative.

Hence x = 4.

Then y = 12 – 8 = 4.

Answer: 4.