Question

A line passing through the center of a rectangle divides into 2 equal partsProve that any line passing through the centre of a rectangle will surely divide the rectangle in 2 equal parts​

Answer 1

Given : line passing through the centre of a rectangle  

To find : Prove that  it will surely divide the rectangle in 2 equal parts​

Solution:

Let say a rectangle  ABCD

and center O

Let say line passing through center cut line AD at M  & BC at N

Draw a line parallel to AB & CD passing through O

intersecting AD at P  & BC  at Q

Compare ΔPOM & QON

PO = QQ   ( as O is center )

∠POM = ∠QON   ( vertically opposite angle )

∠OPM  = ∠ OQN  ( RIght angle )

Hence ΔPOM ≅ QON

=> QN = PM

&  MO = NO

BN = BQ  + QN

=> BN = BC/2  +  QN

DM = DP + PM  =  AD/2  +  PM

BC/2  +  QN = AD/2  +  PM   ( ∵ BC = AD  & QN = PM )

=> BN = DM

Hence CN = AM

Now comparing two trapezium  formed

ABNM  &  CDMN

AB  = CD   ( opposite side of rectangle )

BN = DM   ( shown above)

MN = MN    ( Common)

AM  = CN    ( shown above)

∠B = ∠D

∠C = ∠A

Hence ABNM  ≅  CDMN

Hence any line line passing through the centre of a rectangle will surely divide the rectangle in 2 equal parts​

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Answer 2

A line passing throught the center of reactangle divides into two equal parts ( to prove).

Let the reactangle be ABCD,

AB||CD

and BC||AD.

Let O be the mid point of reactangle ABCD.

then a line MP passes throught the centre of reactangle at O and mid point of BC and AD at M,N .Which divide the reactangle into two equal parts.