A circle has a radius of 17 cm. Two of its chords are parallel and separated by a distance of 23 cm. If the length of one of these chords is 30 cm, find the length of the second chord.
16 cm
The perpendicular from the centre to a chord bisects it, so for a chord of length \(\ell\) at distance \(d\) from the centre, \(d = \sqrt{r^2 - (\ell/2)^2}\).
For the first chord of length 30 cm, half-length \(= 15\) cm.
Its distance from the centre \(= \sqrt{17^2 - 15^2} = \sqrt{289 - 225} = \sqrt{64} = 8\) cm.
The chords are on opposite sides of the centre (since 8 cm plus the second distance must give 23 cm apart), so the second chord's distance \(= 23 - 8 = 15\) cm.
Half of the second chord \(= \sqrt{17^2 - 15^2} = \sqrt{289 - 225} = \sqrt{64} = 8\) cm.
So the second chord \(= 2 \times 8 = 16\) cm.
The length of the second chord is 16 cm.
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