Question

L and M are the mid-points of two equal chords AB and CD and O is the centre of the circle Prove that
(i) $\angle OLM=\angle OML$
(ii) $\angle ALM=\angle CML$

Answer 1

AB and CD are two equal chords of a circle with centre O.L is the mid-point of AB and M is the mid-point of CD

We have to prove

(i) $\angle OLM=\angle OML$

(ii) $\angle ALM=\angle CML$

L is the mid-point of chord AB of circle with centre O

$\Rightarrow OL\perp AB$

Similarly $OM\perp CD$

But Chord AB = Chord CD

$\Rightarrow OL=OM$

(Equal chords are equidistant from the centre )

In $△OLM\Rightarrow OL=OM$

$\Rightarrow \angle OLM=\angle OML$ (Angles opposite to equal sides of a $△$

But $\angle OLA=\angle OMC$....(Each = ${90}^{\circ }$ )

On adding $\angle OLM+\angle OLA=\angle OML+\angle OMC$

$\Rightarrow \angle ALM=\angle CML$