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Q.

In the given figure, O is the centre of a circle, BOA is its diameter and the tangent at the point P meets BA extended at T. If ∠PBO=30⁰ then, which of the following is the measure of ∠PTA?


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a

60⁰

b

30⁰

c

15⁰

d

45⁰ 

answer is B.

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Detailed Solution

Given that, O is the centre of a circle, BOA is its diameter and the tangent at the point P meets BA extended at T such that ∠PBO= 30 °   .
We know that the radius and tangent are perpendicular at their point of contact.
Therefore,
Question ImageWe can observe that a chord passing through the centre is the diameter of the circle.
According to the alternate segment theorem, "The angle measure between a chord of a circle and a tangent through any of the chord's endpoints is equal to the measure of an angle in the alternate segment."
Then, ∠APT=∠ABP= 30 °  .
Consider ∆ABP,
∠PBA+∠BPA+∠BAP= 180 °   [Angle sum property of a triangle] ⇒ 30 ° + 90 ° +∠BAP= 180 °   ⇒∠BAP= 60 °  
Therefore using the Exterior triangle property, we have,
∠BAP=∠APT+∠PTA   ⇒ 60 ° = 30 ° +∠PTA    ⇒∠PTA= 30 °   Hence, the measure of angle PTA is 30 °  .
Therefore, option 2 is correct.
 
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