Question

a trapiziod has vertices A(-2,3) B(4,1) C(-1,-1) and D(2,-2). prove that the line joining the midpoints of the non-parallel sides is the average length of the two parallel sides of the trapaziod.

Answer 1

First, just to check, the slope of AB = -1/3, and so is the slope of CD. So, the non-parallel sides are AC and BD.

|AB| = sqrt(40) = 2sqrt(10)
|CD| = sqrt(10)
(|AB|+|CD|)/2 = 3/2 sqrt(10)

M = midpoint(AC) = (-3/2,1)
N = midpoint(BD) = (3,-1/2)
|MN| = sqrt(81/4 + 9/4) = sqrt(90/4) = 3/2 sqrt(10)