Question

In a circle with centre at point O, chord AB is a side of a square and chord BC is a side of regular hexagon. Then angle AOC is equal to ______.

Options

• 120°

• 150°

• 90°

• none of these

Answer 1

In a circle with centre at point O, chord AB is a side of a square and chord BC is a side of regular hexagon. Then angle AOC is equal to 150°.

Explanation:

• Since chord AB is a side of an inscribed square, it subtends a central angle ∠AOB equal to one-fourth of a full circle:
  `\angle AOB = \frac{360^\circ}{4} = 90^\circ`
• Since chord BC is a side of an inscribed regular hexagon, it subtends a central angle ∠BOC equal to one-sixth of a full circle:
  `\angle BOC = \frac{360^\circ}{6} = 60^\circ`
• The total central angle ∠AOC is the sum of these two adjacent central angles:
  ∠AOC = ∠AOB + ∠BOC
  = 90° + 60°
  = 150°