Question

In the given figure, two tangents PT and QT are drawn to a circle with centre O from an external point T. Prove that \(\angle\) PQT = \(2\angle\) OPQ.

Answer 1

- Let \(PT\) and \(QT\) be two tangents drawn from the external point \(T\) to the circle with center \(O\).

- Since \(PT\) and \(QT\) are tangents, the angle between a tangent and the radius is always \(90^\circ\), so:

\[ \angle OTP = \angle OTQ = 90^\circ \]

- Also, we know that \(\angle PTQ = \angle OTP + \angle OTQ\), which means:

\[ \angle PTQ = 90^\circ + 90^\circ = 180^\circ \]

- Therefore, \(\angle PTQ = 2 \angle OPQ\).