Question

A chord 6 cm long is 1 cm away from the centre of the circle. What is the length of a chord which is 2 cm away from the centre?

Answer 1

Given, a chord 6 cm long is 1 cm away from the centre of the circle.

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.

Let the radius of the circle be $r$.

Then, we must have,

$(\frac{1}{2}\times 6{)}^2=r^2-1^2$

$⟹9=r^2-1$

$⟹r^2=10$

$⟹r=\sqrt{10}$ cm

We shall now find the length of the chord which is 2 cm away from the centre.

Let the length of the chord which is 2 cm away from the centre be $l$ cm.

Then $(\frac{1}{2}\times l{)}^2=r^2-2^2$

$⟹(\frac{1}{2}\times l{)}^2={\sqrt{10}}^2-2^2$

$⟹(\frac{1}{2}\times l{)}^2=6$

$⟹(\frac{1}{2}\times l)=\sqrt{6}$

$⟹l=2\sqrt{6}$ cm