Question

Two concentric circles are of radii 5 cm and 3 cm, respectively. Find the length of the chord of the larger circle that touches the smaller circle.

Answer 1

Given: Two circles have the same centre O and AB is a chord of the larger circle touching the
smaller circle at C; also, OA = 5 cm and OC = 3 cm.
$$\begin{gathered} \mathrm{In}\mathrm{}∆\mathrm{OAC},\mathrm{}{\mathrm{OA}}^2={\mathrm{OC}}^2+{\mathrm{AC}}^2 \\ \therefore {\mathrm{AC}}^2={\mathrm{OA}}^2-{\mathrm{OC}}^2 \\ \Rightarrow {\mathrm{AC}}^2=5^2-3^2 \\ \Rightarrow {\mathrm{AC}}^2=25-9 \\ \Rightarrow {\mathrm{AC}}^2=16 \\ \Rightarrow \mathrm{AC}=4\mathrm{}\mathrm{cm} \\ \therefore \mathrm{AB}=2\mathrm{AC}\mathrm{}\left(\mathrm{since}\mathrm{}\mathrm{perpendicular}\mathrm{}\mathrm{drawn}\mathrm{}\mathrm{from}\mathrm{}\mathrm{the}\mathrm{}\mathrm{centre}\mathrm{}\mathrm{of}\mathrm{}\mathrm{the}\mathrm{}\mathrm{circle} \\ \mathrm{bisects}\mathrm{}\mathrm{the}\mathrm{}\mathrm{chord}\right) \\ \therefore \mathrm{AB}=2\times 4=8\mathrm{}\mathrm{cm} \end{gathered}$$
The length of the chord of the larger circle is 8 cm.