Question

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

Answer 1

Let ABCD is a quadrilateral and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively. i.e., AS = SD, AP = BP, BQ = CQ and CR = DR. We have to show that PR and SQ bisect each other i.e., SO = OQ and PO = OR.
Now, in $\Delta$ ADC, S and R are mid-point of AD and CD.
We know that the line segment joining the mid-points of two sides of a triangle is parallel to the third side. (By mid-point theorem)
$\therefore$ SR || AC and SR = $\frac{1}{2}$ AC ....(i)
Diagonals of a parallelogram bisect each other.
So, SQ and PR bisect each other
Similarly, in $\Delta$ ABC, P and Q are mid - point of AB and BC.
PQ || AC and PQ = $\frac{1}{2}$ AC ( By mid-point theorem) ……(ii)
From Equations (i) and (ii) we get
PQ || RS
And PQ || SR = $\frac{1}{2}$ AC
Therefore, Quadrilateral PQRS is a parallelogram whose diagonals are SQ and PR. Also, we know that diagonals of a parallelogram bisect each other. So, SQ and PR bisect each other