Show that the line segment joining the mid-points of a pair of opposite sides of a parallelogram, divides it into two equal parallelogram.

Given,

ABCD be a parallelogram in which P and Q are mid-points of AB and CD respectively.

Let us construct DG ⊥ AB and let DG = h, where h is the altitude on side AB.

Area of ||gm ABCD = base × height = AB × h

Area of ||gm APQD = AP × h = $\dfrac{AB}{2}$ × h ……(1) [Since P is the mid-point of AB]

The perpendicular distance between two parallel lines is always same everywhere, DG = QR = h.

Area of ||gm PBCQ = PB × h = $\dfrac{AB}{2}$ × h ……(2) [Since P is the mid-point of AB]

From (1) and (2)

Area of ||gm APQD = Area of ||gm PBCQ.

Hence proved, that the line segment joining the mid-points of a pair of opposite sides of a parallelogram divides it into two equal parallelograms.