AB is a chord of a circle with center O. A tangent is drawn at point B, which intersects the extended line AO at point T. If ∠BAT = 21°, then find the value of ∠BTA.

Q.AB is a chord of a circle with center O. A tangent is drawn at point B, which intersects the extended line AO at point T. If ∠BAT = 21°, then find the value of ∠BTA.

In triangle OAB, \(\angle\)OBA = \(\angle\)OAB = \(21^\circ\) [because OA = OB = radius of the circle] \(\therefore \angle\)AOB = \(180^\circ - (21^\circ + 21^\circ) = 180^\circ - 42^\circ = 138^\circ\) \(\therefore \angle\)BOT = \(180^\circ - 138^\circ = 42^\circ\) Now, in triangle BOT: \(\angle\)BOT = \(42^\circ\) \(\angle\)OBT = \(90^\circ\) [because BT is a tangent at point B] \(\therefore \angle\)BTO = \(180^\circ - (42^\circ + 90^\circ) = 180^\circ - 132^\circ = 48^\circ\) \(\therefore \angle\)BTA = \(\angle\)BTO = \(48^\circ\)