\(e.\) Xét \(\Delta AHC\) vuông tại H:
\(AC^2=AH^2+CH^2\left(Pytago\right).\\ \Rightarrow12^2=\left(\dfrac{60}{13}\right)^2+CH^2.\\ \Rightarrow144=\dfrac{3600}{169}+CH^2.\\ \Rightarrow CH=\sqrt{144-\dfrac{3600}{169}}=\dfrac{144}{13}\left(cm\right).\)
Xét \(\Delta ABC\) vuông tại A, đường cao AH:
\(AC^2=CH.BC\) (Hệ thức lượng).
\(144=\dfrac{144}{13}.BC.\\ \Rightarrow BC=13\left(cm\right).\\ BC=CH+BH.\\ \Rightarrow13=\dfrac{144}{13}+BH.\\ \Rightarrow BH=\dfrac{25}{13}\left(cm\right).\)
\(BC^2=AB^2+AC^2\left(Pytago\right).\\\Rightarrow13^2=AB^2+12^2.\\ \Rightarrow AB^2=25.\\ \Rightarrow AB=5\left(cm\right).\)
\(f.\dfrac{HB}{HC}=\dfrac{1}{3}\left(gt\right).\\ \Rightarrow HC=3HB.\)
Xét \(\Delta ABC\) vuông tại A, đường cao AH:
\(AH^2=HB.HC\) (Hệ thức lượng).
\(\Rightarrow\sqrt{12}^2=HB.3HB.\\ \Rightarrow12=3HB^2.\\ \Rightarrow HB^2=4.\\ \Rightarrow HB=2\left(đvđd\right).\\ HC=3HB\left(cmt\right).\\ \Rightarrow HC=3.2=6\left(đcđd\right).\)
\(BC=HB+HC.\\ \Rightarrow BC=2+6=8\left(đvđd\right).\)
