Question : The chord of a circle is equal to its radius. Find the difference between the angle subtended by this chord at the minor arc and the major arc of the circle.
Option 1: 30°
Option 2: 120°
Option 3: 60°
Option 4: 150°
Correct Answer: 120°
Solution : The chord AB is equal to the radius of the circle. OA and OB are the two radii of the circle. AB is the chord of the circle. From $\triangle OAB$, AB = OA = OB = radius of the circle ⇒ $\triangle OAB$ is an equilateral triangle ⇒ $\angle {AOC}=60°$ And $\angle{ACB}=\frac12\angle{AOB}$ ⇒ $\angle{ACB}=\frac12=12×60°=30°$ Now, ACBD is a cyclic quadrilateral, ⇒ $\angle{ADB}+\angle{ADB}=180°$ (Since they are the opposite angles of a cyclic quadrilateral) ⇒ $\angle{ADB}=180°-30°=150°$ ⇒ The angle subtended by the chord at a point on the minor arc and also at a point on the major arc is 150° & 30° respectively. ⇒ Difference = 150° - 30° = 120° Hence, the correct answer is 120°.
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