ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig 8.29). AC is a diagonal. Show that :

(i) SR || AC and SR = 1 /2 AC

(ii) PQ = SR

(iii) PQRS is a parallelogram.

(i) In ∆ACD
 S is the mid-point of AD and R is the mid-point of CD.
SR = 1/2AC and SR || AC …(1)
[By mid-point theorem]

(ii) In ∆ABC, P is the mid-point of AB and Q is the mid-point of BC.
PQ = 1/2AC and PQ || AC …(2)
[By mid-point theorem]

From (1) and (2), we get
PQ = 1/2AC = SR and PQ || AC || SR
⇒ PQ = SR and PQ || SR

(iii) In a quadrilateral PQRS,
PQ = SR and PQ || SR [Proved]
Therefore, PQRS is a parallelogram.