Let ABCD be a trapezoid with midsegment MN, and suppose the diagonals split MN into three segments MP, PQ, and QN. If AB=8 and CD = 14, find the lengths of the midsegment MN and the three segments MP, PQ, and QN.
+128448 See the image below :
The length of the midsegment = (sum of bases ) / 2 = (14 + 8) / 2 = 11
For convenience, let the height of the trapezoid = 8 (it could be any height ....it works out the same)
Note that DEC is similar to triangle BEA
Therefore the height of triangle DEC = (8/14) = of triangle BEA
So...the height of triangle DEC = 8 / (8 + 14) * (8) = (8/22)(8) = 64/22 = 32/11
So the height of triangle QEP = 4 - (32/11) = 12/11
And triangles DEC and QEP are similar
So PQ = (12/32)(8) = 96/32 = 3
And, by symmetery , PQ = QN = (11 - PQ) / 2 = (11 - 3) / 2 = 8/2 = 4
