Question

PQ is tangent to a circle with centre O. If ∠POR = 65°, then m∠PTR is

Options

• 65°

• 58.5°

• 57.5°

• 45°

Answer 1

57.5°

Explanation:

Given:

PQ is a tangent to the circle at point P.

O is the centre of the circle.

∠POR = 65°

PS is the diameter of the circle.

To find: m∠PTR

Calculation:

In ΔORS, OR = OS (radii of the same circle).

So, ∠ORS = ∠OSR (angles opposite to equal sides).

In ΔORS, ∠POR is an exterior angle.

Exterior angle is equal to the sum of two interior opposite angles.

∠POR = ∠ORS + ∠OSR

65° = ∠OSR + ∠OSR

65° = 2∠OSR

∠OSR = 32.5°

Now, in ΔPTS:

∠TPS = 90° (radius OP ⊥ tangent PQ)

∠TPS = 32.5° (calculated above as ∠OSR)

Using angle sum property in ΔPTS:

∠PTS + ∠TPS + ∠TSP = 180°

∠PTR + 90° + 32.5° = 180°

∠PTR + 122.5° + 180°

∠PTR = 180° – 122.5°

∠PTR = 57.5°