Question

A chord 8 cm long is 3 cm away from the centre of the circle A. Calculate the length of a chord that is 4 cm away from the centre of circle A.

Answer 1

Given, a chord 8 cm long is 3 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,

$(\frac{1}{2}\times 8{)}^2=r^2-3^2$

$⟹16=r^2-9$

$⟹r^2=25$

$⟹r=5$ cm

We shall now find the length of the chord which is 4 cm away from the centre.

Let the length of the chord which is 4 cm away from the centre be $l$ cm.

Then $(\frac{1}{2}\times l{)}^2=r^2-4^2$

$⟹(\frac{1}{2}\times l{)}^2=5^2-4^2$

$⟹(\frac{1}{2}\times l{)}^2=9$

$⟹(\frac{1}{2}\times l)=3$

$⟹l=6$ cm