In the given figure, AB is the diameter of the circle, with center O, and AT is the tangent. Calculate the numerical value of x.

In △OBC,

OB = OC (Radius of same circle)

As, angles opposite to equal sides are equal.

∴ ∠OBC = ∠OCB

As, exterior angle is equal to the sum of two opposite interior angles.

∴ ∠COA = ∠OBC + ∠OCB

⇒ ∠COA = 2∠OBC

⇒ 2∠OBC = 64°

⇒ ∠OBC = 32°.

In △ABT,

∠BAT = 90° (∵ Tangent at any point of a circle and the radius through this point are perpendicular to each other.)

⇒ ∠BAT + ∠ABT + ∠ATB = 180° (By angle sum property of triangle)

⇒ 90° + 32° + x° = 180° [∵ ∠ABT and ∠OBC is the same angle]

⇒ x° + 122° = 180°

⇒ x° = 180° - 122°

⇒ x° = 58°.

Hence, x = 58°.