Question

In $△ABC,D,E$ and $F$ are the mid points of $BC,CA$ and $AB$ respectively, then, $BDEF$=________$ABC$

• A. $2$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $\frac{31}{4}$

Answer 1

The correct option is B $\frac{1}{2}$

In $△ABC$,

(I) $E$ and $F$ are the mid-points of side $AC$ and $AB$ respectively.

Therefore, $EF\parallel BC$ and $EF=\frac{1}{2}BC$ (Mid-point theorem)

However, $BD=\frac{1}{2}BC(D$ is the mid-point of $BC)$

Therefore, $BD=EF$ and $BD\parallel EF$

Therefore, $BDEF$ is a parallelogram.

(ii) Using the result obtained above, it can be said that quadrilaterals $BDEF,DCEF,AFDE$ are parallelograms.

We know that diagonal of a parallelogram divides it into two triangles of equal area.

$\therefore$ Area $\left(\Delta BFD\right)=$ Area $\left(\Delta DEF\right)$ (For parallelogram $BDEF$)

Area $\left(\Delta CDE\right)=$ Area ( $△DEF)$ (For parallelogram $DCEF$)

Area $\left(△AFE\right)=$ Area ( $△DEF)$ (For parallelogram $AFDE$)

$\therefore$ Area $\left(\Delta AFE\right)=$ Area $\left(\Delta BFD\right)=$ Area $\left(\Delta CDE\right)=$ Area $\left(\Delta DEF\right)$

Also, Area $\left(\Delta AFE\right)+$ Area $\left(\Delta BDF\right)+$ Area $\left(\Delta CDE\right)+$ Area $\left(\Delta DEF\right)=$ Area $\left(\Delta ABC\right)$

$\Rightarrow$ Area $\left(\Delta DEF\right)+$ Area $\left(\Delta DEF\right)+$ Area $\left(\Delta DEF\right)+$ Area $\left(\Delta DEF\right)=$ Area $\left(\Delta ABC\right)$

$\Rightarrow 4$ Area $\left(\Delta DEF\right)=$ Area $\left(\Delta ABC\right)$

$\Rightarrow$ Area $\left(\Delta DEF\right)=\frac{1}{4}$ Area $\left(△ABC\right)$

(iii) Area (parallelogram $BDEF$) $=$ Area ( $\Delta DEF$ ) + Area ( $△BDF$ )

$\Rightarrow$ Area (parallelogram $BDEF$) $=$ Area ( $\Delta DEF$ ) + Area ( $△DEF$ )

$\Rightarrow$ Area (parallelogram $BDEF$) $=2$ Area ( $\Delta DEF)$

$\Rightarrow$ Area (parallelogram $BDEF$) $=2\times \frac{1}{4}$ Area ( $\Delta ABC)$

$\Rightarrow$ Area (parallelogram $BDEF$) $=\frac{1}{2}$ Area ( $\Delta ABC)$

So, $\text{B}$ is the correct option.