AB and CD are two parallel chords of a circle such that AB = 6 cm and CD = 2AB. Both chords are on the same side of the center of the circle. If the distance between them is equal to one-fourth of the length of the CD, then the diameter of the circle is:
Show answer & explanation
Correct answer: C
Explanation
The correct answer is Option 3 i.e. 6√5 cm.
The perpendicular drawn from the center to the chord of a circle bisects the chord.
Given
AB = 6,
CD = 2AB = 12,
PQ = CD/4 = 3.
OP ⊥ CD and
OQ ⊥ AB
Suppose OP = x
According to the concept:
PD = CD/2 = 6 cm
QB = AB/2 = 3 cm
From Pythagoras theorem in triangle OPD:
r2 = x2 + 62 ………………..(1)
In triangle BOQ:
r2 = (x + 3)2 + 32
r2 = x2 + 18 + 6x ………..(2)
Equating both equations:
x2 + 62 = x2 + 18 + 6x
⇒ 6x = 18
⇒ x = 3
From equation 1:
r2 = 32 + 62
⇒ r2 = 9 + 36 = 45
⇒ r = 3√5 cm
Diameter = 2r = 6√5 cm