# ABCD is a parallelogram with a base of BC. BFE is an straight line cut through the AD in equal 2 part at point F. CDE is another extended straight line that touches the point E. If the area of the parallelogram ABCD is 80 cm^2, DE = CD, and AF = FD, find the area of a) triangle BCE, b) triangle ABF, c) triangle EFC.

We have a parallelogram ABCD, with the point F dividing AD in two equal parts.

Let us take the distance between the sides BC and AD as d.

Now, BC*d = AD*d = 80 cm^2.

The area of the triangle ABF is (1/2)*(AD/2)*d

= 80/4...

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We have a parallelogram ABCD, with the point F dividing AD in two equal parts.

Let us take the distance between the sides BC and AD as d.

Now, BC*d = AD*d = 80 cm^2.

The area of the triangle ABF is (1/2)*(AD/2)*d

= 80/4 = 20 cm^2

Similarly if we take the triangle DFC, the area is (1/2)*(AD/2)*h

= 80/4 = 20 cm^2

The area of the triangle BFC is 80 - 20 - 20 = 40 cm^2.

Now we draw a line EX perpendicular to BC from the point E.

If we consider angle ECX,

sin ECX  =  EX / EC = d / DC

we have DC = (EC /2)

So we get EX = EC*d/DC

=> EC*d*/(EC/2)

=> d* 2

The area of triangle BCE = (1/2)*(BC)*EX

=> (1/2)*BC*d*2

=> BC*d

=> 80 cm^2

The area of the triangle EFC = area of triangle BCE - area of triangle BFC.

We have derived the area of the triangle BFC as 40, so the area of EFC = 80 - 40 = 40 cm^2,

Therefore the area of the triangle BCE is 80 cm^2, of triangle ABF is 20 cm^2 and of triangle EFC is 40 cm^2

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