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Show that the quadrilateral formed by joining the mid points of the sides of a square is also a square

Posted by Sonali Verma 2 years, 4 months ago

CBSE > Class 09 > Mathematics

1 answers

Preeti Dabral 2 years, 4 months ago

Let a square ABCD in which L, M, N & O are the midpoints.
In $\triangle$ AML and $\triangle$ CNO
AM = CO (AB = DC and M and O are the midpoints)
AL = CN (AD = BC and L and N are the midpoints)
$\angle$ MAL = $\angle$ NCO (all angles of a square = 90°)
by AAS criteria
$\triangle$ AML $\cong$ $\triangle$ CNO
$\therefore$ ML = ON (CPCT)
similarly $\triangle$ MBN $\cong$ $\triangle$ LDO
now,
in $\triangle$ AML,
$\angle$ AML = $\angle$ ALM (AM = AL)
= 45°
similarly in $\triangle$ LDO
$\angle$ DLO = 45°
$\therefore$ $\angle$ MLO = 90°
by the properties of SQUARE
All sides are equal and angles are 90°

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CBSE > Class 09 > Mathematics