[latex]BC=10;AC=30 \ \ r=? \ underline{BC^2+AC^2=AB^2 } \ 10^2+30^2=AB^2 \ AB^2=100+900=1000 \ AB= sqrt{1000}= sqrt{100*10}=10 sqrt{10} \ p= frac{AB+BC+AC}{2} \ p= frac{10 sqrt{10}+10+30 }{2}=20+5 sqrt{10} \ S_{ABC}= frac{1}{2}*10*30=5*30=150 \ \ S_{ABC}=pr=(20+5 sqrt{10})r \ \ (20+5 sqrt{10})r=150iff r= frac{150}{20+5 sqrt{10} } = frac{30}{4+ sqrt{10} } \ r= frac{30(4- sqrt{10}) }{16-10} =5(4- sqrt{10}) \ underline{r=5(4- sqrt{10}) } [/latex]