In triangle ACD, If BE || CD, and BC = AB = 3, AE = 4 and CD = 10, what is the area of trapezoid BEDC?

Solution(By Examveda Team)

Since BE||CD, it implies that the triangle ABE and ACD are similar figures.

For triangle ABE,
AB=3
AE=4
BE =?

For triangle ACD,
AC=AB+BC = 3+3 = 6
AD = AE+ED = 4+?
CD =10

Since ABE and ACD are similar figures and AC = 2 AB (6=2*3), we can deduce the unknown sides, as the sides of ABE and ACD will follow the ratio of 1:2

For triangle ABE,
AB=3
AE=4
BE =(1/2)CD = (1/2)10 = 5

For triangle ACD,
AC=AB+BC = 3+3 = 6
AD = 2AE = 2*4 = 8
CD=10

Now that we know all sides, Area(BCDE) = Area(ACD) – Area(ABC)

Area of a triangle = sqrt(s(s-a)(s-b)(s-c)), where s=(a+b+c)/2 and abc are lengths of the sides.

Area(ACD) = sqrt(12*6*2*4) = 24
Area(ABC) = sqrt(6*3*2*1) = 6

Area(BCDE) = 24-6 = 18