Question

ABCD is a rectangle . P and Q are points on sides AD and AB respectively . Prove that APOQ is a rectangle and find the ar (APOQ): ar ( ABCD ) when it is given that BR = BC/4 and DS = DC/4

Answer 1

Given In the rectangle ABCD, P,Q,R and S are points on sides AD,AB,BC and CD, respectively such that AQ = 1/4 AB, AP = 1/4 ADBR = 1/4 BC and DS = 1/4 CD

To prove: 
(1) APOQ is a rectangle 
(2) ar (APOQ) : ar (ABCD) = ?

the proof for this is 
(1) As, AB || CD and AB = CD (opposite sides of rectangle ABCD)

=> AQ || DS and  AQ = DS [1/4 AB = 1/4 CD, given]
but this is a pair of opposite sides of quadrilateral ADSQ
So, ADSQ is a parallelogram
=> AD || QS (opposite sides of parallelogram ADSQ)
=> AP || QO -- (i)
Also, AD || BC and AD = BC (oppostie sides of rectanlge ABCD)
=> AP || BR and AP = BR [1/4 AD = 1/4 BC, given]

but this is a pair of opposite sides of the quadrilateral APRB
so, APRB is a parallelogram
=> AB || PR (opposite sides of parallelogram ADSQ)
=> AQ || PO -- (ii)

APOQ is a parallelogram
Since ∠ A = 90 degrees (each angle of rectangle is 90 degrees)
but this is an angle of parallelogram APOQ
so APOQ is a rectangle 
(2)
ar (APOQ) = AQ x AP
= 1/4 AB x 1/4 AD
= 1/16 (AB x AD)
= 1/16 ar (ABCD)
=> ar (APOQ) / ar (ABCD) = 1/16
ar (APOQ) ar (ABCD) = 1:16