Question

In $△ABC$, if D and E are midpoints of BC and AD respectively such that $ar\left(△AEC\right)=4c{m}^2$, then $ar\left(△BEC\right)$ =

• A. $4c{m}^2$
• B. $6c{m}^2$
• C. $8c{m}^2$
  
• D. $12c{m}^2$
  

Answer 1

The correct option is C

$8c{m}^2$

In $△ABC,$ D and E are the mid points of BC and AD
join BE and CE are $\left(△AEC\right)=4c{m}^2$

In $△ABC,$
$\because$ AD is the median of BC
$\therefore ar\left(△ABD\right)=ar\left(△ACD\right)$
Similarly in $△EBC,$ ED is the median
$\therefore ar\left(△EBD\right)=ar\left(△ECD\right)$
and in $△ADC,CE$ is the median
$\therefore ar\left(△EDC\right)=ar\left(△AEC\right)=4c{m}^2$

$\therefore ar\left(△BEC\right)=2\times ar\left(△EDC\right)=2\times 4=8c{m}^2$