57
In the given figure, $PQ$ is a chord of the circle centered at $O$. $PT$ is a tangent to the circle at $P$. If $\angle QPT = 55^\circ$, then find $\angle PRQ$.
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$\angle QPT = 55^\circ$
Since $OP \perp PT$ (radius is perpendicular to tangent at point of contact)
$\angle OPQ = 90^\circ - \angle QPT = 90^\circ - 55^\circ = 35^\circ$
In $\triangle OPQ$, $OP = OQ$ (radii of same circle)
$\Rightarrow \angle OQP = \angle OPQ = 35^\circ$
$\angle POQ = 180^\circ - (\angle OPQ + \angle OQP) = 180^\circ - (35^\circ + 35^\circ) = 180^\circ - 70^\circ = 110^\circ$
The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
$\angle PRQ = \frac{1}{2} \times \text{reflex } \angle POQ$
Reflex $\angle POQ = 360^\circ - 110^\circ = 250^\circ$
Hence $\angle PRQ = \frac{1}{2} \times 250^\circ = 125^\circ$
Since $OP \perp PT$ (radius is perpendicular to tangent at point of contact)
$\angle OPQ = 90^\circ - \angle QPT = 90^\circ - 55^\circ = 35^\circ$
In $\triangle OPQ$, $OP = OQ$ (radii of same circle)
$\Rightarrow \angle OQP = \angle OPQ = 35^\circ$
$\angle POQ = 180^\circ - (\angle OPQ + \angle OQP) = 180^\circ - (35^\circ + 35^\circ) = 180^\circ - 70^\circ = 110^\circ$
The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
$\angle PRQ = \frac{1}{2} \times \text{reflex } \angle POQ$
Reflex $\angle POQ = 360^\circ - 110^\circ = 250^\circ$
Hence $\angle PRQ = \frac{1}{2} \times 250^\circ = 125^\circ$