The median and the height of an isosceles trapezoid are equal in length. One diagonal of the trapezoid has length 12. What is the area of the trapezoid?
See the following image ( not to scale)
Call the length of the longer base, AB = P and the length of the shorter base, GF = Q
Then AF = Q + ( P - Q) / 2 = (P + Q) / 2 = the median length = M
So....since the median = the height, we have right triangle AFD such that
AF^2 + FD^2 = AD^2 where AF, FD = M and AD = 12
M^2 + M^2 = 12^2
2M^2 = 144
M^2 = 72
M = 6sqrt (2) = height
So....the area = (1/2) height * (sum of the bases) = height * (sum of bases)/2 =
height * median =
(6sqrt (2) ) ( 6sqrt (2)) = 36 * 2 = 72 units^2