Three externally tangent circles $C_1,C_2,C_3$ have centers $O_1,O_2,O_3$ and radii $r_1=2$ cm,$r_2=1$ cm,$r_3$. The angle $O_1O_3O_2$ is $90^\circ$,as shown. Find $r_3$ in cm.
External tangency gives center distances $O_1O_2=3$, $O_1O_3=2+r_3$ and $O_2O_3=1+r_3$. Apply Pythagoras at $O_3$:$$(2+r_3)^2+(1+r_3)^2=3^2.$$ Hence $r_3^2+3r_3-2=0$. Its positive root is $r_3=(-3+\sqrt{17})/2$ cm;the other root is negative and inadmissible. **A**.
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