a quadrilateral has vertices at the points(-4,2) , (2,6), (8,5), (9,-7),show that the mid points of the sides of this quadrilateral are the vertices of a parallelogram.
The mid point of sides AB,BC,CD,DA are P,Q,R,S respectively.
And the mid-point formula is {(x1 + x2)/2 , (y1 + y2)/2}
So P is the midpoint of side AB
Hence P = {(-4+2)/2, (6+2)/2} = {-1, 4}
Q is the mid point of side BC
So Q = {(8+2)/2, (6+5)/2} = {5, 11/2}
R is the mid-point of CD
So R = {(8+9)/2, (5-7)/2} = {17/2, -1}
S is the mid-point of AD
So S = {(-4+9)/2, (2+ 7)/2} = {5/2, -5/2}
For the quadrilateral PQRS to be a parallelogram, the opposite sides are parallel, hence the slope of PQ = SR and slope of RQ = PS
So Now for finding the slope we have the formula m = ,
So slope of PQ =
Slope of RS =
So slope of PQ = RS
And slope of RQ =
Slope of PS =
Hence slope of RQ = PS
Hence PQRS is parallelogram