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In the adjoining figure, ABCD is a quadrilateral in which P, Q, R and S are mid-points of AB, BC, CD

and DA respectively. AC is its diagonal. Show that

(i) SR || AC and SR = ½ AC

(ii) PQ = SR

(iii) PQRS is a parallelogram.

Answer:

It is given that

In quadrilateral ABCD

P, Q, R, and S are the mid-points of sides AB, BC, CD, and DA

AC is the diagonal

To find:

(i) SR || AC and SR = ½ AC

(ii) PQ = SR

(iii) PQRS is a parallelogram

Proof:

(i) In Δ ADC

S and R are the mid-points of AD and DC

SR || AC and SR = ½ AC….. (1) Using the mid-point theorem

(ii) In Δ ABC

P and Q are the midpoints of AB and BC

PQ || AC and PQ = ½ AC ….. (2)

Using equation (1) and (2)

PQ = SR and PQ || SR

(iii) PQ = SR and PQ || SR

Hence, PQRS is a parallelogram.

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