Question

In the given figure, AB ∥ SQ ∥ DC and AD ∥ PR ∥ BC. If the area of quadrilateral ABCD is 24 square units, find the area of quadrilateral PQRS.

Answer 1

Let SQ and PR intersect at point O.
Now,
DC || SQ and RP || BCC
⇒ RC || OQ and RO || QC
⇒ QuadrilateralROQC is a parallelogram.
Similarly,
Quadrilateral ROSD, APOS and POQB are parallelograms.
ΔROQ and parallelogram ROQC are on the same base and between the same parallel lines.

∴ A(ΔROQ) = `(1)/(2)` x A(||^gm ROQC)       .....(i)

Similarly,
A(ΔPOQ) = `(1)/(2)` x A(||^gm POQB)       .....(ii)

A(ΔPOS) = `(1)/(2)` x A(||^gm APOS)       .....(iii)

A(ΔSOR) = `(1)/(2)` x A(||^gm ROSD)       .....(iv)

Adding equations (i), (ii), (iii) and (iv), we get
A(ΔROQ) + A(ΔPOQ) + A(ΔPOS) + A(ΔSOR)

= `(1)/(2)`[A(||^gm ROQC) + A(||^gm POQB) + A(||^gm APOS) + A(||^gm ROSD)]

⇒ A(||^gm PQRS) 

= `(1)/(2)` x A(||^gm ABCD) 

= `(1)/(2) xx 24`
= 12 sq. units.