Question

A chord QR subtends an angle of 105° at the centre O of the circle. The measure of ∠RQP is

Options

• `(75^circ)/2`

• `(105^circ)/2`

• 75°

• 15°

Answer 1

`bb((105^circ)/2)`

Explanation:

Given:

∠QOR = 105°

PQ is a tangent at Q.

In ΔOQR:

OQ = OR   ...(Radii of the same circle)

∠OQR = ∠ORQ   ...(Angles opposite to equal sides)

Using angle sum property in ΔOQR:

∠OQR + ∠ORQ + ∠QOR = 180°

2∠OQR + 105° = 180°

2∠OQR = 180° – 105°

2∠OQR = 75°

∠OQR = `(75^circ)/2`

Since PQ is a tangent at Q, OQ ⊥ PQ:

∠OQP = 90°

From the figure, ∠RQP is the angle between chord QR and tangent PQ.

By Alternate Segment Theorem, the angle between a chord and a tangent is equal to the angle subtended by the chord in the alternate segment. 

∠RQP = ∠OQP – ∠OQR

`∠RQP = 90^circ - (75^circ)/2`

`∠RQP = (180^circ - 75^circ)/2`

`∠RQP = (105^circ)/2`