Question

A segment of a circle is constructed with chord of length 5 cm and the angle subtented at a point on the circumference is ${60}^{\circ }$. Find the angles of the triangle formed by the centre of the circle and the chord.

• A. ${120}^{\circ }$,${50}^{\circ }$,${70}^{\circ }$
• B. ${60}^{\circ }$,${30}^{\circ }$,${90}^{\circ }$
• C. ${60}^{\circ }$,${60}^{\circ }$,${60}^{\circ }$
• D. ${120}^{\circ }$,${30}^{\circ }$,${30}^{\circ }$

Answer 1

The correct option is D ${120}^{\circ }$,${30}^{\circ }$,${30}^{\circ }$

Given
Length of the chord AB = 5 cm
Angle subtended by an arc at the centre is twice the angle at the circumference.
$\Rightarrow \angle BOA={120}^{\circ }$
and also $\angle OBA=\angle OAB=x$
[ $\because OA=OB$ (radii of same circle)]


Sum of all angles in a triangle = 180°
⇒ x + x + 120° = 180°
⇒ 2x + 120° = 180°
⇒ 2x = 180° - 120° = 60°
⇒ $x=\frac{60}{2}=30°$
⇒$\angle OBA=\angle OAB=30°$