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 {(

*x*/2 , (

_{1}+ x_{2})*y*)/2}

_{1}+ y_{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 = $\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$,

So slope of PQ = $\frac{{\displaystyle \frac{11}{2}}-4}{5-(-1)}=\frac{1}{4}$

Slope of RS = $\frac{{\displaystyle \frac{-5}{2}}-(-1)}{{\displaystyle \frac{5}{2}}-{\displaystyle \frac{17}{2}}}=\frac{1}{4}$

So slope of PQ = RS

And slope of RQ = $\frac{-1-{\displaystyle \frac{11}{2}}}{5-{\displaystyle \frac{17}{2}}}=\frac{{\displaystyle \frac{-13}{2}}}{{\displaystyle \frac{-7}{2}}}=\frac{13}{7}$

Slope of PS = $\frac{{\displaystyle \frac{-5}{2}}-4}{-1-{\displaystyle \frac{5}{2}}}=\frac{13}{7}$

Hence slope of RQ = PS

Hence PQRS is parallelogram

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