In the given figure, is a diameter of a circle with center and

If calculate

(i) (ii)

(iii) (iv)

Also, show that is an equilateral triangle.

(i)

ABCD is a cyclic quadrilateral.

∴ ∠BAD + ∠BCD = 180°

⇒ ∠BAD + 120° = 180°

⇒ ∠BAD = 60°

(ii)

∠BDA = 90°[Angle in a semi-circle]

In triangle ABD,

∠ABD + ∠BDA + ∠BAD = 180°[Sum of angles of triangle]

⇒ ∠ABD + 90° + 60° = 180°

⇒ ∠ABD + 150° = 180°

⇒ ∠ABD = 30°

(iii)

OD = OA [Radius]

∴ ∠OAD = ∠ODA = ∠BAD = 180°

∴∠ODB = 90° - ∠ODA

⇒ ∠ODB = 90° - 60°

⇒ ∠ODB = 30°

(iv)

∠ADC = ∠ADB + ∠CDB

⇒ ∠ADC = 90° + 30°

⇒ ∠ADC = 120°

In triangle AOD,

∠AOD + ∠OAD + ∠ODA = 180°[Sum of angles of triangle]

⇒ ∠AOD + 60° + 60° = 180°

⇒ ∠AOD + 120° = 180°

⇒ ∠AOD = 60°

∴ Triangle AOD is an equilateral triangle.