Question

ABCD is a trapezoid with sides AB parallel to CD, where AB = 50, CD = 20. E is a point on the side AB with the property, that the segment DE divides the given trapezoid into two parts of equal area (see figure). Calculate the length AE.

Answer 1

AE is 35 units in length.

One of the two shapes that DE splits the trapezoid into is a triangle.  Since the two sections of the trapezoid have an equal area, this means that the area of the triangle is 1/2 of the area of the trapezoid.  Using the formulas for the area of a triangle and the area of a trapezoid we get:

1/2bh = 1/2(1/2(B+b)(h))

The base of the triangle, AE, is unknown.  We do know that AE + EB = 50; let x be EB.  That means that AE = 50-x.

B, the "big base" of the trapezoid, is 50.  b in the trapezoid, the little base, is 20.  Using all of this we now have:

1/2(50-x)h = 1/2(1/2(50+20)(h))
1/2(50-x)h = 1/2(1/2(70)h)
1/2(50-x)h=1/2(35)h

Since we have multiplied by 1/2 and h on both sides, we can divide by both of them at the same time to cancel.  This will give us:
(50-x)=35

Subtract 50 from both sides:
50-x=50 = 35-50
-x = -15
x = 15

This means that EB is 15; thus AE = 50-15 =35.