Question

Two parallel chords 10 cm and 24 cm long are drawn on the same side of the centre of a circle of radius 13 cm. Find the distance between the chords.

Answer 1

We need to find PQ in the above figure.

Since perpendicular from the centre to a chord bisects the chord, we must have AP = BP = 12 cm and CQ = DQ = 5 cm.

We know, in a circle, the square of half the length of a chord is the difference of the squares of the radius and the perpendicular distance of the chord from the centre of the circle.

So, $A{P}^2=O{A}^2-O{P}^2$

$⟹O{P}^2=O{A}^2-A{P}^2$

$⟹O{P}^2={13}^2-{12}^2=169-144=25=5^2$

$⟹OP=5$ cm

Similarly,

$OQ=\sqrt{O{C}^2-C{Q}^2}=\sqrt{{13}^2-5^2}=12$ cm

Now, $PQ=OQ-OP=12\text{cm}-5\text{cm}=7\text{cm}$

Therefore the distance between the chords is 7 cm.