Question

ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA ( see the given figure). AC is a diagonal. Show that:

PQ = SR

Answer 1

In $\Delta$ADC, S and R are the mid-points of sides AD and CD respectively. In a triangle, the line segment joining the mid-points of any two sides of the triangle is parallel to the third side and is half of it.
SR || AC and SR = $\frac{1}{2}$AC …….(1)

In $\Delta$ABC, P and Q are mid - points of sides AB and BC respectively. Therefore, by using mid-point theorem,
PQ || AC and PQ = $\frac{1}{2}$AC .........(2)
Using equations (1) and (2) , we obtain
PQ || SR and PQ = SR …..(3)
$\therefore$ PQ = SR