-Qua D kẻ đường thẳng song song BI cắt AC tại F.
-Xét △ABC: AD là tia p/g của \(\widehat{BAC}\) (gt)
\(\Rightarrow\dfrac{BD}{CD}=\dfrac{AB}{AC}\) (định lí đường phân giác trong tam giác)
\(\Rightarrow\dfrac{BD}{CD}=\dfrac{10}{35}=\dfrac{2}{7}\)
-Có: \(AE=\dfrac{3}{4}AD\) (gt) ; \(AE+ED=AD\)
\(\Rightarrow\dfrac{3}{4}AD+ED=AD\)
\(\Rightarrow ED=\dfrac{1}{4}AD\)
\(\Rightarrow\dfrac{AE}{ED}=\dfrac{\dfrac{3}{4}AD}{\dfrac{1}{4}AD}=3\)
-Xét △AIF: EI//DF.
\(\Rightarrow\dfrac{AI}{IF}=\dfrac{AE}{ED}=3\) (định lí Ta-let) (1) \(\Rightarrow IF=\dfrac{1}{3}AI\)
-Xét △IBC: DF//BI.
\(\Rightarrow\dfrac{IF}{CF}=\dfrac{BD}{CD}=\dfrac{2}{7}\) (định lí Ta-let) (2)
-Từ (1), (2) suy ra:
\(\dfrac{AI}{IF}.\dfrac{IF}{CF}=3.\dfrac{2}{7}=\dfrac{6}{7}\)
\(\Rightarrow\dfrac{AI}{CF}=\dfrac{6}{7}\)
\(\Rightarrow CF=\dfrac{7}{6}AI\)
*\(AI+IF+CF=AC\)
\(\Rightarrow AI+\dfrac{7}{6}AI+\dfrac{1}{3}AI=35\)
\(\Rightarrow\dfrac{5}{2}AI=35\)
\(\Rightarrow AI=14\left(cm\right)\)
