Question

Out of the two concentric circles , the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle . Find the radius of the inner circle .

Answer 1

Let the centre of the two concentric circles be O.
CD is the tangent to the inner circle.
OP joins the centre of the circle to the tangent at the point of contant.
OP $\perp$ CD and OP bisects CD.
Thus, PD = 4 cm
In ∆OPD,
$$\begin{gathered} {\mathrm{OP}}^2+{\mathrm{PD}}^2={\mathrm{OD}}^2 \\ \Rightarrow {\mathrm{OP}}^2={\mathrm{OD}}^2-{\mathrm{PD}}^2 \\ \Rightarrow {\mathrm{OP}}^2={\left(5\right)}^2-{\left(4\right)}^2=25-16=9 \\ \Rightarrow \mathrm{OP}=3\mathrm{cm} \end{gathered}$$
Hence, the radius of the inner circle = 3 cm.