Question

In the figure, $ED\parallel QO$ and $DF\parallel OR$. Prove that,
$EF\parallel QR$

Answer 1

In $\Delta PQR,DE\parallel OQ$

In $\Delta PRO,DF\parallel OR$

Since if line deacon parallel to one side of triangle,

intersects the other two sides in distrinct points, then it divides the other two side in the same ratio.

$\Rightarrow \frac{PE}{EQ}=\frac{PD}{DO}\cdots \left(1\right)$

$\Rightarrow \frac{PF}{FR}=\frac{PD}{DO}\cdots \left(2\right)$

From (1) and (2) $\frac{PF}{FR}=\frac{PE}{EQ}$

i.e., line EF divides the triangle PQR in the same ratio.

Using the fact that if a line divides any two sides of a triangle in the same ratio then the line is parallel to the third side

$\therefore EF\parallel QR.$

Hence proved.