Question

Two concentric circles are of radii $5$ cm and $3$cm. Find the length of chord of the larger circle which touches the smaller circle.

Answer 1

Let two circles have the same centre $O$ and $AB$ is a chord of the larger circle touching the smaller circle at $C$.

$OA=5cm$ and $OC=3cm$

Now,

As we know that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Therefore,

In $△OAC$,

${OA}^2={OC}^2+{AC}^2\left(\text{By pythagoras theorem}\right)$

$\Rightarrow {AC}^2={OA}^2-{OC}^2$

$\Rightarrow {AC}^2={\left(5\right)}^2-{\left(3\right)}^2$

$\Rightarrow AC=\sqrt{25-9}=\sqrt{16}=4cm$

Since perpendicular drawn from the centre of circle bisects the chord.

Therefore,

$AB=2AC$

$\Rightarrow AB=2\times 4=8cm$

Hence the length of the chord is $8cm$.