Question:
In the given figure, O is the centre of the circle AB is a chord and AT is the tangent at A. If ∠AOB = 100∘ then ∠BAT is equal to
(a) 40∘
(b) 50∘
(c) 90∘
(d) 100∘
Solution:
Given: AO and BO are the radius of the circle
Since, AO = BO
∴ △AOB is an isosceles triangle.
Now, in △AOB
∠AOB + ∠OBA + ∠OAB = 180∘ (Angle sum property of triangle)
⇒ 100∘ + ∠OAB + ∠OAB = 180∘ (∠OBA = ∠OAB)
⇒ 2∠OAB = 80∘
⇒ ∠OAB = 40∘
We know that the radius and tangent are perperpendular at their point of contact
∵∠OAT = 90∘
⇒ ∠OAB + ∠BAT = 90∘
⇒ ∠BAT = 90∘ − 40∘ = 50∘
Hence, the correct answer is option (b).