Question

In the given figure , A and B are centres of two circles touching each other at M. Line AC and line BD are tangents. If AD = 6 cm and BC = 9 cm then find the length of AC and BD.

Answer 1

AM = AD, MB = BC
AB = AM + MB = 6 + 9 = 15 cm


$\mathrm{Here},\angle \mathrm{ACB}=90°\left(\mathrm{Tangent}\mathrm{is}\mathrm{perpendicular}\mathrm{to}\mathrm{the}\mathrm{radius}\right)\mathrm{So},\mathrm{in}\mathrm{right}\Delta \mathrm{ABC},\mathrm{we}\mathrm{have}:\mathrm{AC}=\sqrt{{\mathrm{AB}}^2-{\mathrm{BC}}^2}=\sqrt{{15}^2-9^2}=\sqrt{225-81}=\sqrt{144}=12\mathrm{cm}$

$\mathrm{Similarly},\angle \mathrm{ADB}=90°\left(\mathrm{Tangent}\mathrm{is}\mathrm{perpendicular}\mathrm{to}\mathrm{the}\mathrm{radius}\right)\mathrm{So},\mathrm{in}\mathrm{right}\Delta \mathrm{ABD},\mathrm{we}\mathrm{have}:\mathrm{BD}=\sqrt{{\mathrm{AB}}^2-{\mathrm{AD}}^2}=\sqrt{{15}^2-6^2}=\sqrt{225-36}=\sqrt{189}\mathrm{cm}$