A chord \(QR\) subtends an angle of \(105^\circ\) at the centre \(O\) of the circle. The measure of \(\angle RQP\) is
Show Hint
By the Alternate Segment Theorem, the angle between the tangent \(QP\) and chord \(QR\) is equal to the inscribed angle subtended by the same chord \(QR\) on the major arc.
The inscribed angle is always half of the central angle subtended by the same arc:
\[ \angle \text{inscribed} = \frac{\angle \text{centre}}{2} = \frac{105^\circ}{2} \]
This provides the answer in just one step!
Step 1: Understand the setup. A chord QR subtends an angle of \(105^\circ\) at the centre O of the circle. PQ is a tangent to the circle at point Q. We need to find \(\angle RQP\). Step 2: Find the angle at the centre. The central angle for minor arc QR is \(\angle QOR = 105^\circ\). Step 3: Use the tangent-chord angle theorem. The angle between a tangent and a chord at the point of tangency equals half the central angle subtended by that chord on the same side. Step 4: Apply the theorem. \(\angle RQP = \frac{1}{2} \times \angle QOR = \frac{1}{2} \times 105^\circ = \frac{105^\circ}{2}\). Step 5: Confirm using the alternate segment theorem. The tangent-chord angle equals the inscribed angle in the alternate segment. The inscribed angle subtended by arc QR in the alternate segment \(= \frac{105^\circ}{2}\), confirming our result. Step 6: Select the correct option. \(\angle RQP = \frac{105^\circ}{2}\), which is option 2. \[ \boxed{\dfrac{105^\circ}{2}} \]
