Question

Given A = (7, 1, 3), C = (4, -2, 3), B = (5, 1, 2), and D = (6, m, n), find m and n so that segments BD and AC have the same midpoint. Is ABCD a parallelogram?
Please help!

Answer 1

The midpoint of AC = ( (7+4)/2 , (1-2)/2 , (3+3)/2) = (11/2 , -1/2 , 3)
midpoint of BD = ( (5+6)/2 , (1+m)/2, (2+n)/2) = (11/2, (1+m)/2, (2+n)/2)
so (1+m)/2 = -1/2
1+m = -1
m = -2

find n in the same way.

Find the slopes of AB and CD, the should be same
Find the slopes of AD and BC, they should be the same if ABCD is a parallelogram.