Question

The radius of a circle is equal to the chord of a circle. The chord's angle at a position on the major arc and at a position on the minor arc are

• A. ${30}^{\circ },{150}^{\circ }$
• B. ${150}^{\circ },{30}^{\circ }$
• C. ${60}^{\circ },{120}^{\circ }$
• D. ${120}^{\circ },{60}^{\circ }$

Answer 1

The correct option is A ${30}^{\circ },{150}^{\circ }$

Given that: Chord $\text{AB}$ is equal to the radius of the circle.

In the $△\text{OAB}$,
$\text{AB = OA = OB =}$Radius
∴ ΔOAB is an equilateral triangle.
Therefore, each interior angle of this triangle will be of ${60}^{\circ }$.
The angle made by the chord at the circumference of the circle will be half of the angle made by the same chord at the center of the circle.

$\therefore \angle \text{ACB}=\frac{\angle \text{AOB}}{2}$
$=\frac{{60}^{\circ }}{2}$
$={30}^{\circ }$

In cyclic quadrilateral $\text{ABCD}$
$\angle \text{ACB}+\angle \text{ADB}={180}^{\circ }$
${30}^{\circ }+\angle \text{ADB}={180}^{\circ }$
$\angle \text{ADB}={180}^{\circ }-{30}^{\circ }$
$\angle \text{ADB}={150}^{\circ }$