Question

State whether the following statement is true or false.

In the figure $\angle BED=\angle BDE$ and $E$ divides $BC$ in the ratio $2:1$ then $AF\times BE=2AD\times CF$.

• A. True
• B. False

Answer 1

The correct option is A True

Through $C$ draw $CG\parallel FD.$

Now, In $△AFD,$

$CG\parallel FD$

$\Rightarrow$ $\frac{AC}{CF}=\frac{AG}{GD}$ [ BPT ]

$\Rightarrow$ $\frac{AC}{CF}+1=\frac{AG}{GD}+1$

$\Rightarrow$ $\frac{AC+CF}{CF}=\frac{AG+GD}{GD}$

$\Rightarrow$ $\frac{AF}{CF}=\frac{AD}{GD}$ ------ ( 1 )

In $△BCG,$

Since, $DE\parallel GC,$ then

$\Rightarrow$ $\frac{BE}{EC}=\frac{BD}{GD}$ [ BPT ]

$\Rightarrow$ $\frac{2}{1}=\frac{BD}{GD}$ [ As, $BE:EC=2:1$ ]

$\Rightarrow$ $BD=2GD$ ----- ( 2 )

In $△BDE,$

$\Rightarrow$ $\angle BED=\angle BDE$ [ Given ]

$\Rightarrow$ $BD=BE$ [ Sides opposite to equal angles are equal ] ----- ( 3 )

From ( 2 ) and ( 3 ), we get

$\Rightarrow$ $2GD=BE$

$\Rightarrow$ $GD=\frac{BE}{2}$ ----- ( 4 )

Substituting the value of $GD$ from ( 4 ) in ( 1 ), we get

$\Rightarrow$ $\frac{AF}{CF}=\frac{AD}{\frac{BE}{2}}$

$\Rightarrow$ $\frac{AF}{CF}=\frac{2AD}{BE}$

$\Rightarrow$ $AF\times BE=2AD\times CF$