Question

Equal chords AB and CD of a circle with centre O, cut at right angles at E. If M and N are the mid - points of AB and CD respectively, then OMEN is a __.

Answer 1

Given: Equal chords AB and CD of a circle with centre O cut at right angles at E. M and N are the middle points of AB and CD respectively.

$\angle$OMB = $\angle$OND = ${90}^{\circ }$

$\angle$OME = $\angle$ONE = ${90}^{\circ }$..........(1)

OM = ON ............(2) [Since equal chords of a circle are equidistant from the centre]

In $\Delta$OME and $\Delta$ONE

OM = ON [From (2)]

$\angle$OME = $\angle$ONE [Each equal to ${90}^{\circ }$]

and, OE = OE [Common]

$\Delta$OME $\cong$ $\Delta$ONE [RHS theorem of congruence]
$\therefore$ ME = NE [C.P.C.T]

In quadrilateral OMEN, OM = ON , ME = NE and $\angle$OME = $\angle$ONE = ${90}^{\circ }$

Hence, it is a square.