Question : AB and BC are two chords of a circle with centre O. Both chords are on either side of the centre O. Point A and point C are connected to the centre O, such that $\angle B A O=36^{\circ}$ and $\angle B C O=48^{\circ}$. What is the degree measure of the angle subtended by the minor arc AC at the centre O?
Option 1: 136°
Option 2: 144°
Option 3: 120°
Option 4: 168°
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Correct Answer: 168°
Solution :
In $\triangle$BOC,
OB = OC (since they are radii)
$\angle$OBC = $\angle$OCB = 48°
$\angle$BOC = 180° – (48° + 48°) = 84°
In $\triangle$AOB,
OA = OB (since they are radii)
$\angle$OAB = $\angle$OBA = 36°
⇒ $\angle$AOB = 180° – (36° + 36°) = 108°
$\angle$AOC = 360° – ($\angle$AOB + $\angle$BOC)
⇒ $\angle$AOC = 360° – (84° + 108°)
$\therefore \angle$AOC = 168°
Hence, the correct answer is 168°.
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