Question

Prove that if a line is drawn parallel to one side of a triangle intersecting the other two side,then it divides the two sides in the same ratio.

Answer 1

Given $DE\parallel BCIn△ADEand△ABC\angle DAE=\angle BAC\left(commonangle\right)\angle ADE=\angle ABC[\because DE\parallel BC\therefore correspondinganglesareequal]\angle AED=\angle ACB\therefore △ADE\sim △ABCHence\frac{AD}{AB}=\frac{AE}{AC}(Insimilartriangle,correspondingsidesareinsameratio)$

$\because \frac{AD}{AB}=\frac{AE}{AC}\Rightarrow \frac{AB}{AD}=\frac{AC}{AE}\Rightarrow \frac{AB}{AD}-1=\frac{AC}{AE}-1\Rightarrow \frac{AB-AD}{AD}=\frac{AC-AE}{AE}\Rightarrow \frac{DB}{AD}=\frac{EC}{AE}\Rightarrow \frac{AD}{DB}=\frac{AE}{EC}$

It is proved that if a line is drawn parallel to one side of a triangle intersecting the other two side, then it divides the two sides in the same ratio.