Question

Two circles of radii $8cm$ and $3cm$ have a direct common tangent of length $10cm$ . Find the distance between their centres, upto two places of decimal.

Answer 1

$O{O}^{\prime }$ = distance between centres

$OA=3cm$ $OB=8cm$

$AB=10cm$

clearly $\Delta OAM\sim \Delta OBM$

$\therefore \frac{3}{8}=\frac{OA}{OB}=\frac{AM}{BM}=\frac{OM}{O^{\prime }M}$

$\frac{AM}{BM}=\frac{3}{8}$ and $AB=10$

$\therefore AM=\frac{3}{11}\times 10;BM=\frac{8}{11}\times 10$

$AM=\frac{30}{11},BM=\frac{80}{11}$

$OM=\sqrt{O{A}^2+A{M}^2}=\sqrt{9+\frac{900}{121}}=4.054$

$O^{\prime }M=\sqrt{O^{\prime }{B}^2+B{M}^2}=\sqrt{64+\frac{6400}{121}}=10.811$

$\therefore OM+OM=00=14.86cm$