Question

In the figure given below (not drawn to scale), A, B and C are three points on a circle with centre O. The chord BA is extended to a point T such that CT becomes a tangent to the circle at point C. If $\angle$ATC = 30${}^{\circ }$ and $\angle$ACT = 50${}^{\circ }$, then the angle $\angle$BOA is :

• A. 100°
• B. 150°
• C. 80°
• D. not possible to determine

Answer 1

The correct option is A 100°

$\angle$ CAT= 100${}^{\circ }$

Therefore, $\angle$BAC= 80${}^{\circ }$

$\angle$OCT= 90${}^{\circ }$

$\angle$OCA = 40${}^{\circ }$
Since triangle OCA is an isosceles triangle, $\angle$ OAC = 40 ${}^{\circ }$
Simillarly, $\angle$ OBA = 40 ${}^{\circ }$
Therefore, $\angle$ AOB = 100 ${}^{\circ }$