Question

Given two concentric circles of radii 5 and 3, find the length of a chord of larger circle which touches the smaller one. If BD = 5, find BC

Answer 1

$$\begin{gathered} \angle \mathrm{OPB}=90°(\mathrm{Tangent}\mathrm{is}\mathrm{perpendicular}\mathrm{to}\mathrm{radius}) \\ \mathrm{In}\mathrm{right}\mathrm{triangle}\mathrm{OPB},\mathrm{w}e\mathrm{have}: \\ \mathrm{PB}=\sqrt{{\mathrm{OB}}^2-{\mathrm{OP}}^2}=\sqrt{5^2-3^2}=\sqrt{25-9}=\sqrt{16}=4\mathrm{units} \end{gathered}$$
$\mathrm{AB}=2\mathrm{BP}=8\mathrm{units}$
Since BD is a secant and BP is a tangent and by secant-tangent theorem, we have:
$$\begin{gathered} \mathrm{BC}\times \mathrm{BD}={\mathrm{PB}}^2 \\ \Rightarrow \mathrm{BC}\times 5=4^2 \\ \Rightarrow \mathrm{BC}=\frac{16}{5}=3.2\mathrm{units} \end{gathered}$$