Question

If two intersecting chords of a circle make equal angles with diameter passing through their point of intersection, prove that the chords are equal.

Answer 1

Given, $\angle AEQ=\angle DEQ$

$OL\perp AB,OM\perp DC$

In $△OLE$ and $△OME$

$\angle OLE=\angle OME$ (Both ${90}^o$)

$\angle OEL=\angle OEM$ (given, $\angle AEQ=\angle DEQ$)

$OE=OE$ (common)

$\therefore △OLE\cong △OME$ (AAS congruence)

$OL=OM$ (CPCT)

$OL$ is the distance of chord $AB$ from centre and $OM$ is the

distance of chord $CD$ from centre.

Since, $OL=OM$

chord $AB$ and $CD$ are equidistant from centre

Hence, $AB=CD$ (chords at equal distance from centre)