Question

Show that the line segments joining the mid-points of opposite sides of a quadrilateral bisects each other.

Answer 1

Let ABCD is the quadrilateral and P, Q, R, S are mid points of the sides AB, BC, CD, DA respectively.

Join DB to form triangle ABD.

$$\begin{gathered} \frac{\mathrm{AS}}{\mathrm{SD}}=\frac{\mathrm{AP}}{\mathrm{PB}} \\ \Rightarrow \mathrm{SP}\left|\right|\mathrm{DB}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}\mathrm{SP}=\frac{1}{2}\hspace{0.17em}\mathrm{DB} \end{gathered}$$

In triangle BCD

$$\begin{gathered} \frac{\mathrm{CR}}{\mathrm{RD}}=\frac{\mathrm{CQ}}{\mathrm{QB}} \\ \Rightarrow \mathrm{RQ}\left|\right|\mathrm{DB}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}\mathrm{RQ}=\frac{1}{2}\hspace{0.17em}\hspace{0.17em}\mathrm{DB} \end{gathered}$$

In quadrilateral PQRS,
SP = RQ and SP || RQ
∴ PQRS is a parallelogram.

Diagonals of a parallelogram bisect each other.

∴ PR and QS bisect each other.