NCERT Solutions
Class 9 Maths
Quadrilaterals

Ex.8.2 Q.1
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:
(1) SR || AC and SR =
(2) PQ = SR
(3) PQRS is a parallelogram.
(1) In ∆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.
So, SR || AC and SR = .................(1)
(2) In ∆ABC, P and Q are mid-points of sides AB and BC respectively.
Therefore, by using mid-point theorem,
PQ || AC and PQ = AC ..............(2)
Using equations 1 and 2, we obtain
PQ || SR and PQ = SR ……………. (3)
So, PQ = SR
(3) From equation 3, we obtained PQ || SR and PQ = SR
Clearly, one pair of opposite sides of quadrilateral PQRS is parallel and equal.
Hence, PQRS is a parallelogram.