2015-11-11 04:54:12
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
2015-11-11 11:01:37

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)