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# A Chord of Length 24 Cm is at a Distance of 5 Cm from the Centre of the Circle. Find the Length of the Chord of the Same Circle Which is at a Distance of 12 Cm from the Centre. - ICSE Class 10 - Mathematics

ConceptChord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof)

#### Question

A chord of length 24 cm is at a distance of 5 cm from the centre of the circle. Find the length of the chord of the same circle which is at a distance of 12 cm from the centre.

#### Solution

Let AB be the chord of length 24 cm and O be the centre of the circle.
Let OC be the perpendicular drawn from O to AB.
We know, that the perpendicular to a chord, from the centre of a circle, bisects the chord.

∴ AC = CB = 12 cm
In ∆OCA,

OA2 = OC2 + AC2  (By Pythagoras theorem)
= (5)2 + (12)2 = 169
⇒  OA = 13 cm
∴ radius of the circle = 13 cm

Let A'B' be new chord at a distance of 12 cm from the centre.
∴ (OA')2 = (OC')2 + (A'C')2
⇒ (A'C')2 = (13)2 - (12)2 = 25
∴ A'C' = 5 cm

Hence, length of the new chord = 2 × 5 = 10 cm

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Solution A Chord of Length 24 Cm is at a Distance of 5 Cm from the Centre of the Circle. Find the Length of the Chord of the Same Circle Which is at a Distance of 12 Cm from the Centre. Concept: Chord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof).
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