Question

In the given figure, PQ is a tangent to a circle with centre O. A is the point of contact. If ∠PAB = 67^∘, then the measure of ∠AQB is
(a) 73^∘
(b) 64^∘
(c) 53^∘
(d) 44^∘

Answer 1

We know that a chord passing through the centre is the diameter of the circle.
∵∠BAC = 90^∘ (Angle in a semi circle is 90^∘)
By using alternate segment theorem
We have ∠PAB = ∠ACB = 67^∘
Now, In △ABC
∠ABC + ∠ACB + ∠BAC = 180^∘ [Angle sum property of a triangle]
⇒ ∠ABC + 67^∘ + 90^∘ = 180^∘
⇒ ∠ABC= 23^∘
Now, ∠BAQ = 180^∘ − ∠PAB [Linear pair angles]
= 180^∘ − 67^∘
= 113^∘
Now, In △ABQ
∠ABQ + ∠AQB + ∠BAQ = 180^∘ [Angle sum property of a triangle]
⇒ 23^∘ + ∠AQB + 113^∘ = 180^∘
⇒ ∠AQB = 44^∘
Hence, the correct answer is option (d).