Question

A chord $AB$ of a circle subtends an angle of ${50}^o$ at a point on the circumference of the circle. The acute angle between the tangents at $A$ and $B$ has magnitude

• A. ${80}^o$
• B. ${65}^o$
• C. ${75}^o$
• D. ${85}^o$
• E. ${82}^o$

Answer 1

The correct option is A ${80}^o$

Solution:

Given that:

$\angle ACB={50}^{\circ }$

To find:

$\angle ATB=?$

Solution:

$\angle AOB=2\angle ACB$ (Angle subtended by a chord at the centre is twice of the angle subtended at the circumference.)

$\angle AOB=2\times {50}^{\circ }={100}^{\circ }$

In quadrilateral $AOBT,$

$\angle OAB+\angle AOB+\angle OBT+\angle ATB={360}^{\circ }$ $(OA\perp AT,OB\perp BT)$

or, ${90}^{\circ }+{100}^{\circ }+{90}^{\circ }+\angle ATB={360}^{\circ }$

or, $\angle ATB={80}^{\circ }$

Hence, A is the correct option.