Two chords of lengths a metre and b metre subtend angles 60 deg and 90 deg at the centre of the circ...........

Quantitative Aptitude ( CCS) Geometry

Two chords of lengths a metre and b metre subtend angles 60° and 90° at the centre of the circle respectively. Which of the following is true ?

b = √2a

a = √2b

a = 2b

b = 2a

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Explanation:

OA = OB = OC = OD = r units (radii)
AB = a metre and CD = b metre
∠AOB = 60° and ∠COD = 90°
In ΔCOD, By pythagoras theorem
CD² = OC² + OD²
b² = r² + r² = 2r² ...(i)
In ΔAOB,
OA = OB
∴ ∠ABO = ∠OAB
∠AOB + ∠ABO + ∠OAB = 180°
60° + ∠OAB + ∠OAB = 180°
2∠OAB = 180° – 60° = 120°
∠OAB = 60° = ∠ABO
∴ ΔAOB is an equilateral triangle.
OA = OB = AB ⇒ a = r
From equation (i),
b = √2r
b = √2a