SOLUTION: Write a coordinate proof to show that the segments connecting the midpoints of any quadrilateral form a parallelogram. I think it is simpler to use vectors but not sure if I have t

Algebra ->  Geometry-proofs -> SOLUTION: Write a coordinate proof to show that the segments connecting the midpoints of any quadrilateral form a parallelogram. I think it is simpler to use vectors but not sure if I have t      Log On


   

Question 1206128: Write a coordinate proof to show that the segments connecting the midpoints of any quadrilateral form a parallelogram. I think it is simpler to use vectors but not sure if I have to use midpoint formulas. Thank you.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let the coordinates of the vertices of the quadrilateral be

A(2a,2b)
B(2c,2d)
C(2e,2f)
D(2g,2h)

The midpoints are

AB: P(a+c,b+d)
BC: Q(c+e,d+f)
CD: R(e+g,f+h)
DA: S(g+a,h+b)

Opposite sides PQ and RS have the same slope:
PQ: %28%28b%2Bd%29-%28d%2Bf%29%29%2F%28%28a%2Bc%29-%28c%2Be%29%29=%28b-f%29%2F%28a-e%29
RS:

And opposite side QR and PS have the same slope:
QR: %28%28d%2Bf%29-%28f%2Bh%29%29%2F%28%28c%2Be%29-%28e%2Bg%29%29=%28d-h%29%2F%28c-g%29
PS: %28%28b%2Bd%29-%28h%2Bb%29%29%2F%28%28a%2Bc%29-%28g%2Ba%29%29=%28d-h%29%2F%28c-g%29

Both pairs of opposite sides are parallel, making the quadrilateral a parallelogram.