Cho tam giác ABC có: AB = 4cm, AC = 5cm, BC = 6cm. Tỉ số diện tích các tam giác DIE và ABC là:

* Hướng dẫn giải

Ta có:

 \(\begin{array}{l} \frac{{AD}}{{AB}} = \frac{{DC}}{{BC}}(t/c)\\ \Rightarrow \frac{{AD}}{4} = \frac{{DC}}{6} = \frac{{AD + DC}}{4} = \frac{5}{{10}} = \frac{1}{2}\\ \Rightarrow AD = 4.\frac{1}{2} = 2,DC = 6.\frac{1}{2} = 3 \end{array}\)

Suy ra:

\(\begin{array}{l} \frac{{DI}}{{IB}} = \frac{{DC}}{{CB}} = \frac{3}{6} = \frac{1}{2} \Rightarrow \frac{{DI}}{{DB}} = \frac{1}{3}\\ \frac{{BE}}{{EA}} = \frac{{BC}}{{AC}} = \frac{6}{5} \Rightarrow \frac{{BE}}{{BA}} = \frac{6}{{11}}\\ \frac{{AD}}{{DC}} = \frac{2}{3} \Rightarrow \frac{{AD}}{{AC}} = \frac{2}{5} \end{array}\)

Suy ra

\({S_{DIE}} = \frac{1}{3}.\frac{6}{{11}}.\frac{2}{5} = \frac{4}{{55}}{S_{ABC}}\)

Vậy:

\(\frac{{{S_{DIE}}}}{{{S_{ABC}}}} = \frac{4}{{55}}\)