\(\dfrac{HB}{HC}=\dfrac{2}{5}\Rightarrow\dfrac{HB}{2}=\dfrac{HC}{5}=\dfrac{HB.HC}{2.5}=\dfrac{AH^2}{10}=\dfrac{256}{10}=\dfrac{128}{5}\)
\(\Rightarrow HB=\dfrac{128}{5}.2=\dfrac{256}{5}\left(cm\right);HC=\dfrac{128}{5}.5=128\left(cm\right)\)
\(\Rightarrow BC=HB+HC=\dfrac{256}{5}+128=\dfrac{896}{5}\left(cm\right)\)
\(AC^2=AH^2+HC^2=256+\left(\dfrac{256}{2}\right)^2=256\left(1+\dfrac{256}{4}\right)\Rightarrow AC=16\sqrt[]{1+\dfrac{256}{4}}=16\sqrt[]{\dfrac{260}{4}}=16.\dfrac{1}{2}.2\sqrt[]{65}=16\sqrt[]{65}\left(cm\right)\)
\(AB^2=AH^2+BH^2=256+\left(\dfrac{256}{5}\right)^2=256\left(1+\dfrac{256}{25}\right)\Rightarrow AB=16\sqrt[]{1+\dfrac{256}{25}}=\dfrac{16}{5}\sqrt[]{281}\left(cm\right)\)
Chu vi tam giác ABC là : \(AB+AC+BC\)
\(=\dfrac{16}{5}\sqrt[]{281}+16\sqrt[]{65}+\dfrac{896}{5}\)
\(=16\left(\dfrac{1}{5}\sqrt[]{281}+\sqrt[]{65}+\dfrac{56}{5}\right)\)
\(=16\left(\sqrt[]{65}+\dfrac{56+\sqrt[]{281}}{5}\right)\left(cm\right)\)
