Given trapezoid abcd with bases ab and cd, draw diagonals ac and bd. let e be the midpoint of ac and f the midpoint of bd. prove that e and f lie on the mid segment of the trapazoid if ab = 10 and dc = 22 find the lenght of ef

Consider the picture.

Let MN be the midsegment of the trapezoid.

That is M is the midpoint of AD, N is the midpoint of BC.

Being the midsegment of the trapezoid, MN is parallel to the bases.

Let O and K be the intersections of the diagonals with the midsegment.

MN//AB, so MO//AB, and since M is the midpoint of DA, O must be the midpoint of DB, 

Similarly we prove that K is the midpoint of CA.

Thus O is F and K is E.

O and K lie on the midsegment MN, so F and E lie on the midsegment.

MO is a midsegment of triangle ABD so |MO|=1/2 |AB|=1/2 * 10=5

MK is a midsegment of triangle ADC, so |MK|=1/2 * |DC|=1/2 * 22=11

|OK|=|MK|-|MO|=11-5=6 (units)