Circle $C$ has radius 6 cm. How many square centimeters are in the area of the largest possible inscribed triangle having one side as a diameter of circle $C$?
I recall that the area of a triangle is given by $\frac{1}{2}bh$, where $b$ is the base and $h$ is the height. Since the base of the triangle is a diameter of the circle, its length is $2r = 12$ cm. To maximize the height of the triangle, I want to choose the other two vertices of the triangle such that they are as far away from the diameter as possible. I realize that this means the other two vertices must lie on the circle, and they must be symmetric about the diameter. This means the triangle is actually a right triangle, and the height is equal to the radius of the circle, which is $6$ cm. Therefore, the area of the triangle is $\frac{1}{2} \times 12 \times 6 = 36$ square centimeters. 