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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