1 Attachment(s)

what is the aera of the circle with center E?

ok so the problem wants me to find the area of the circle with center E. it was pretty easy to figure out that the diameter of the small circle was 6. once i figure out the diameter i can use that relationship to say that the area of the circle with center A and center C is 36pi. but this is where im stuck. im not sure how im supposed to use these relationships to figure out the area of the circle with center E. any thoughts or help would be greatly appreciated.

Attachment 24534

Re: what is the aera of the circle with center E?

$\displaystyle BC = 3$, so $\displaystyle AC = 6$.

It is easy to show that triangle ACD is equilateral (since the side lengths are all radii of either circle w/ center A or C). Using some 30-60-90 triangles, $\displaystyle BD = 3 \sqrt{3}$, so $\displaystyle ED = 2BD = 6 \sqrt{3}$.

ED is a radius of the circle with center E, so the area of this circle is

$\displaystyle \pi (6 \sqrt{3})^2 = 108 \pi$, answer choice (C).