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  • In the given figure, PA and PB are the tangent segments to a circle with centre O. Show that the points A, O, B and P are concyclic.

In the given figure, PA and PB are the tangent segments to a circle with centre O. Show that the points A, O, B and P are concyclic.

Answers (1)

best_answer

Here, $\mathrm{OA}=\mathrm{OB}$
And $\mathrm{OA} \perp \mathrm{AP}, \mathrm{OA} \perp \mathrm{BP}$ (Since tangents drawn from an external point are perpendicular to the radius at the point of contact)

$
\begin{aligned}
& \therefore \angle \mathrm{OAP}=90^{\circ}, \angle \mathrm{OBP}=90^{\circ} \\
& \therefore \angle \mathrm{OAP}+\angle \mathrm{OBP}=90^{\circ}+90^{\circ}=180^{\circ} \\
& \therefore \angle \mathrm{AOB}+\angle \mathrm{APB}=180^{\circ}\left(\text { Since }, \angle \mathrm{OAP}+\angle \mathrm{OBP}+\angle \mathrm{AOB}+\angle \mathrm{APB}=360^{\circ}\right)
\end{aligned}
$


Sum of opposite angle of a quadrilateral is $180^{\circ}$.
Hence A, O, B and P are concyclic.

Posted by

Vishal kumar

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