smc

Question

The diameter AB of a circle of radius 2 is extended to a point D outside the circle so that BD=3. Point E is chosen so that ED=5 and line ED is perpendicular to line AD. Segment AE intersects the circle at a point C between A and E. What is the area of ABC?

Accepted Answer

FIGURE Let O be the center of the circle. Note that EC + CA = EA = AD^2 + DE^2 = (2+2+3)^2 + 5^2 = 74. However, by Power of a Point, (EC)(EC + CA) = EO^2 - R^2 = (2+3)^2 + 5^2 - 2^2 = 25 + 25 - 4 = 46 EC = 4674, so AC = 74 - 4674 = 2874. Now BC = AB^2 - AC^2 = 4^2 - 28^274 = 16 74 - 28^274 = 1184 - 78474 = 2074. Since ACB = 90^, [ABC] = 12 BC AC = 12 2074 2874 = 14037.