Question

BC is a tangent to the circle with centre O. OD is radius of the circle. If ∠DOC = 100°, ∠B is equal to:

Options

• 50°

• 60°

• 40°

• 70°

Answer 1

40°

Explanation:

From figure,

OD = OA (Radius of same circle)

In △OAD,

∠ODA = ∠OAD = x (let) (As angles opposite to equal sides are equal)

Since, exterior angle in a triangle is equal to the sum of two opposite interior angles.

∴ ∠DOC = ∠ODA + ∠OAD

⇒ 100° = 2x

⇒ x = `(100°)/2`

⇒ x = 50°

⇒ ∠OAD = 50°

We know that,

Tangent at any point of a circle and the radius through this point are perpendicular to each other.

∴ ∠BCA = 90°

In △ABC,

By angle sum property of triangle,

⇒ ∠ABC + ∠BCA + ∠CAB = 180°

⇒ ∠ABC + ∠BCA + ∠OAD = 180° [∵ From figure, ∠CAB = ∠OAD]

⇒ ∠ABC + 90° + 50° = 180°

⇒ ∠ABC + 140° = 180°

⇒ ∠ABC = 180° − 140° = 40°