Question

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio

Answer 1

Given that,$XY||BC$

$ABC$ is a triangle.

To prove;

$\begin{array}{c}\frac{AX}{AB}=\frac{AY}{AC}\\ XY||BC\\ \angle AXY=\angle ABC\\ \angle AYX=\angle ACB\end{array}$ [Corresponding triangle]

In $\Delta AXY$ and $\Delta ABC$

$\begin{array}{c}\angle X=\angle B\\ \angle Y=\angle C\end{array}$
By AA Rule, $\Delta AXY\sim \Delta ABC$ [Similarly]

$\therefore \frac{AX}{AB}=\frac{AY}{AC}$