Question

$PQ$ and $QR$ are two equal chords of a circle. A diameter of the circle is drawn through $Q$. Prove that the diameter bisect $\angle PQR$.

Answer 1

Proving $△OMQ\cong △ONQ$, to prove diameter bisect $\angle PQR$

Let us assume that $QT$ be the diameter
Then, from question it is given that $PQ=QR$
So, $OM=ON$
Now, consider the $△OMQ$ and $△ONQ$,
$\angle OMQ=\angle ONQ$ [both angles are equal to ${90}^{\circ }$ ]
$OM=ON$
[from question it is given that equal chords]
$OQ=OQ$ [common side for both triangles]
Therefore, $△OMQ\cong △ONQ$ [By RHS]
So, $\angle OQM=\angle OQN$ [CPCT]
Hence $QT$ i.e. diameter of the circle bisects $\angle PQR$