Question

0. (-2,4), (4,8), (10,7) and (11,-5) are the vertices of a quadrilateral. Show that the quadrilateral, obtained on joining the mid-points of its sides, is a parallelogram.

Answer 1

Let A(-2,4), B(4,8), C(10,7), D(11,-5) be the vertices of a quadrilateral.

Let E, F, G, H be the mid points of the sides AB, BC, DC and AD respectively.

Then,

Coordinates of E are : $(1,6)$

Coordinates of F are : $(7,\frac{15}{2})$

Coordinated of G are : $(\frac{21}{2},1)$

Coordinates of H are : $(\frac{9}{2},\frac{-1}{2})$

Now slope of EF = $\frac{1}{4}$

Slope of GH = $\frac{1}{4}$

Slope of EH = $-\frac{13}{7}$

Slope of FG = $-\frac{13}{7}$

Hence, opposite sides are parallel to each other.

Also, $EF=GH$ and $EH=FG$

So, $EFGH$ is a parallelogram.