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.

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Solution

## 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. $\mathrm{In}\mathrm{}∆\mathrm{OAC},\mathrm{}{\mathrm{OA}}^{2}={\mathrm{OC}}^{2}+{\mathrm{AC}}^{2}\phantom{\rule{0ex}{0ex}}\therefore {\mathrm{AC}}^{2}={\mathrm{OA}}^{2}-{\mathrm{OC}}^{2}\phantom{\rule{0ex}{0ex}}⇒{\mathrm{AC}}^{2}={5}^{2}-{3}^{2}\phantom{\rule{0ex}{0ex}}⇒{\mathrm{AC}}^{2}=25-9\phantom{\rule{0ex}{0ex}}⇒{\mathrm{AC}}^{2}=16\phantom{\rule{0ex}{0ex}}⇒\mathrm{AC}=4\mathrm{}\mathrm{cm}\phantom{\rule{0ex}{0ex}}\therefore \mathrm{AB}=2\mathrm{AC}\mathrm{}\left(\mathrm{since}\mathrm{}\mathrm{perpendicular}\mathrm{}\mathrm{drawn}\mathrm{}\mathrm{from}\mathrm{}\mathrm{the}\mathrm{}\mathrm{centre}\mathrm{}\mathrm{of}\mathrm{}\mathrm{the}\mathrm{}\mathrm{circle}\phantom{\rule{0ex}{0ex}}\mathrm{bisects}\mathrm{}\mathrm{the}\mathrm{}\mathrm{chord}\right)\phantom{\rule{0ex}{0ex}}\therefore \mathrm{AB}=2×4=8\mathrm{}\mathrm{cm}$ The length of the chord of the larger circle is 8 cm.

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