\(\Delta ABC\)vuông đường cao AH:
\(\hept{\begin{cases}AB^2=BH.BC\\AC^2=CH.BC\end{cases}\Rightarrow\frac{AB^2}{AC^2}=\frac{BH.BC}{CH.BC}=\frac{BH}{CH}}\)
\(\Leftrightarrow\frac{BH}{CH}=\frac{AB^2}{AC^2}=\left(\frac{AB}{AC}\right)^2\)
Vì AD là đường phân giác \(\Delta ABC\)(gt);
\(\frac{BD}{DC}=\frac{AB}{AC}=\frac{51}{68}=\frac{3}{4}\)
\(\Rightarrow\left(\frac{AB}{AC}\right)^2=\left(\frac{3}{4}\right)^2=\frac{9}{16}\)
\(\Rightarrow\frac{BH}{CH}=\frac{9}{14}\)
Áp dụng tính chất dãy tỉ số bằng nhau:
\(\frac{BH}{9}=\frac{CH}{16}=\frac{BH+CH}{9+16}=\frac{BC}{25}=\frac{BD+CD}{25}=\frac{119}{25}\)
\(\Rightarrow BH=\frac{9.119}{25}=42,84cm\)
\(\Rightarrow CH=\frac{16.119}{25}=76,16cm\)
