Question

For the trapezoid, ABCD E and F are the midpoints of

AC
and
BD
respectively.
Find the length of the segment
EF
, if AB = a, BC = b, CD = c, AD = d.

Answer 1

EF = (d − b) / 2

Explanation:

Let's say that G is the intersection of the trapezoid's diagonals.

Triangle GBC is similar to triangle GDA, so we can write a proportion:

GD / GB = AD / BC

GD / GB = d / b

GD = (d / b) GB

Next, F is the midpoint of BD, so BF equals FD.

BF = FD

GB + GF = GD − GF

2GF = GD − GB

2GF = (d / b) GB − GB

2GF = ((d − b) / b) GB

GF / GB = (d − b) / (2b)

Finally, triangle GBC is similar to triangle GFE, so we can write another proportion:

GF / GB = EF / BC

(d − b) / (2b) = EF / b

EF = (d − b) / 2