Question

In the given figure, two concentric circles are drawn. If the radius of an outer circle = 13 cm and the length of the chord of an outer circle which touches an inner circle = 24 cm then, find the radius of inner circle.

• A. 5 cm
• B. 4 cm
• C. 6 cm
• D. 5.5 cm

Answer 1

The correct option is A 5 cm

Given - Radius of outer circle = 13 cm and PQ = 24 cm. PQ is the chord of the outer circle which touches the inner circle at L. Join OL and OP.

By Theorem- In two concentric circles, the chord of the bigger circle, that touches the smaller circle is bisected at the point of contact with the smaller circle.

$\therefore LP=\frac{1}{2}PQ$

$\therefore LP=\frac{1}{2}\times 24=12cm$
so,
LP = 12 cm and OP = 13 cm (Radius of outer circle)

By Theorem- The tangent at any point of a circle is perpendicular to the radius through the point of contact.
so, $\angle$OLP = ${90}^{\circ }$
In right $\Delta$ OLP, applying pythagoras theorem,

$O{P}^2=O{L}^2+L{P}^2$

$\Rightarrow (13{)}^2=(OL{)}^2+{12}^2$

$\Rightarrow O{L}^2=169-144$

$\Rightarrow O{L}^2=25$

$\Rightarrow OL=\sqrt{25}=5cm$

$\therefore$ Radius of inner circle (i.e $OL$) = $5cm$