The conversation between human and AI assistant.
What are all distinct triangles having one side of length 6 with the other 2 sides being integers and the perimeter numerically equal to area?
Why is that the only one?
One way to see it is to use the fact that the area of a triangle with sides a, b, and c is given by the formula A = √(s(s-a)(s-b)(s-c)), where s is the semiperimeter of the triangle, which is half the perimeter. In this case, the semiperimeter is 8 + b, where b is the length of the third side. So we want to find integer solutions to the equation 8 + b = √((8+b)(b)(2(b+8))), which simplifies to b^2 + 16b - 64 = 0. This equation has only one integer solution, which is b = 10. Therefore, the third side is 6, 8, or 10, and only the triangle with sides 6, 8, and 10 satisfies the given conditions.