Question

Equal chords AB and CD of a circle with centre O, cut at right angles at E. If M and N are the midpoints of AB and CD, respectively, then prove that OMEN is a square.

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.

To prove: OMEN is a square.

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

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

$OE=OE$ [Common]

$\Delta OME\cong \Delta ONE$ [RHS theorem of congruence]

$ME=NE$ [C.P.C.T]

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

Hence, it is a square.