Question

ABCD is a rectangle. P and Q are points on sides AD and AB respectively. Show that APOQ ia a rectangle and find ar(APOQ):ar(ABCD),when it is given that BR=1/4 BC and DS=1/4CD.

Answer 1

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

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

Proof:
(1)
As, AB∥CD and AB=CD (Opposite sides of rectangle ABCD)
⇒AQ∥DS and
AQ=DS [1/4AB=1/4CD, 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 (Opposite sides of rectangle ABCD)
⇒AP∥BR and
AP=BR [1/4AD=14B/C, given]
But this is a pair of opposite sides of quadrilateral APRB
So, APRB is a parallelogram
⇒AB∥PR (Opposite sides of parallelogram ADSQ)
⇒AQ∥PO .....(ii)

From (i) and (ii),
APOQ is a parallelogram

Since,
∠A=90. (angle of rectangle is 90°)
but this is an angle of parallelogram APOQ
Hence,
APOQ is a rectangle


(2)
ar(APOQ)=AQ×AP
=1/4AB×1/4AD
=1/16(AB×AD)
=1/16ar(ABCD)

⇒ar(APOQ)ar(ABCD)=1/16

∴ ar(APOQ):ar(ABCD)=1:16





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Answer 2

Answer:

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