prove that any line passing through the centre of rectangle divide rectangle vin 2 equal parts
Answers
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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