inscribed triangle in a circle

[maria08]

The question is as follows:

Circle C has a radius of 2.

Column A
The greatest possible area of any triangle that can be drawn completely within the circle C.

Column B
8

I tried to use awareness or intuition, or whatever one may call this, and my guess is that Column B is bigger but I cannot use reasoning to prove it.

anyhelp would be great.

Maria

[tommywallach]

Hey Maria,

Well, the circle has an area of pi * r^2 = 4pi, or about 12.5

Now, if a triangle had base 4 and height 4 (the diameter of the circle), it's area would be 1/2 * 4 * 4 = 8

But our triangle can't have both a height and a base of 4. If it had a base of 4 (across the whole diameter), it could only have a height of 2 (try drawing it). So the triangle could never be as big as 8, so the answer is B.

Hope that helps!

-t