We have triangle $\triangle ABC$ where $AB = AC$ and $AD$ is an altitude. Meanwhile, $E$ is a point on $AC$ such that $AB \parallel DE.$ If $BC = 12$ and the area of $\triangle ABC$ is $180,$ what is the area of $ABDE$?
See the following image......
Let A = (6,30) B = (0, 0) C = ( 12,0) D = (6,0) and AD = 30
Since ED is parallel to AB.....then ang;e EDC = angle ABC
And angle ACB = angle ECD
So triangle ABC will be similar to triangle EDC
And the base of EDC = DC is 1/2 of base BC....so
The area of triangle EDC = Area of triangle ABC * (scale factor of EDC to ABC)^2 =
!80 * (1/2)^2 = 180 * (1/4) = 45 units^2
So....[ ABDE] = [ ABC ] - [EDC] = 180 - 45 = 135 units^2