In a circle with centre O, a chord AB subtends an angle of 80° at the centre. A tangent at B meets the extension of AB at T. Find the measure of ∠ ATB.

In a circle with centre $O$ , a chord $A B$ subtends an angle of $8 0^{\circ}$ at the centre. A tangent at $B$ meets the extension of $A B$ at $T$ . Find the measure of $∠ A T B$ .

hard 5 marks Class 10 · Mathematics · Circles

Answer

∠ A T B = 14 0^{\circ}

The correct answer is $∠ A T B = 14 0^{\circ}$ — Therefore $∠ A T B$ , which is the angle between chord $A B$ (extended as $A T$ ) and the tangent at $B$ , is $14 0^{\circ}$ .

Step-by-step solution

The angle between a chord and a tangent through one end of the chord equals the angle in the alternate segment.
The alternate segment is the part of the circle opposite the chord; its subtended angle at any point on the circle equals half the central angle of the opposite arc.
The opposite arc of $A B$ has central angle $36 0^{\circ} - 8 0^{\circ} = 28 0^{\circ}$ .
Hence the angle in the alternate segment is $\frac{1}{2} \times 28 0^{\circ} = 14 0^{\circ}$ .
Therefore $∠ A T B$ , which is the angle between chord $A B$ (extended as $A T$ ) and the tangent at $B$ , is $14 0^{\circ}$ .

Common mistake: Taking the angle as

\frac{8 0 ^{\circ}}{2} = 4 0^{\circ}

instead of using the opposite arc

Concept tested: Angle between chord and tangent

Published 25 August 2026 · How this solution was prepared and checked

In a circle with centre O, a chord AB subtends an angle of 80∘ at the centre. A tangent at B meets the extension of AB at T. Find the measure of ∠ATB.

Step-by-step solution

Related questions

In a circle with centre $O$ , a chord $A B$ subtends an angle of $8 0^{\circ}$ at the centre. A tangent at $B$ meets the extension of $A B$ at $T$ . Find the measure of $∠ A T B$ .