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Question
Two parallel chords are drawn in a circle of diameter 30.0 cm. The length of one chord is 24.0 cm and the distance between the two chords is 21.0 cm;
find the length of another chord.
Solution
Since the distance between the chords is greater than the radius of the circle (15 cm), so the chords will be on the opposite sides of the center.
Let O be the center of the circle and AB and CD be the two parallel chords such that AB = 24 cm.
Let the length of the CD be 2x cm.
Drop OE and OF perpendicular on AB and CD from the center O.
OE ⊥ AB and OF ⊥ CD
∴ OE bisects AB and OF bisects CD. ...(Perpendicular drawn from the center of a circle to a chord bisects it.)
⇒ AE = `24/2`
= 12 cm;
CF = `(2x)/2`
= x cm
In right ΔOAE,
OA2 = OE2 + AE2
⇒ OE2 = OA2 - AE2
= 152 - 122
= 81
∴ OE = 9 cm
∴ OF = EF - OE
= 21 - 9
= 12 cm
In right ΔOCF,
OC2 = OF2 + CF2
⇒ x2 = OC2 - OF2
= 152 - 122
= 81
∴ x = 9 cm
Hence, length of chord CD = 2x
= 2 × 9
= 18 cm.
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