Question

ABC is a triangle in which E is mid-point of median AD. Which of the following is not true?

• A. area of $\Delta BED=\frac{1}{4}$ area of $\Delta ABC$
• B. area of $\Delta BED$ = area of $\Delta BAE$
• C. area of $\Delta ABD$ = area of $\Delta ADC$
• D. area of $\Delta BED=\frac{1}{2}$ area of $\Delta ABC$

Answer 1

The correct option is D

area of $\Delta BED=\frac{1}{2}$ area of $\Delta ABC$

In $\Delta ABC$, AD is the median.
$\therefore ar\left(\Delta ABD\right)=ar\left(\Delta ADC\right)$
$\Rightarrow ar\left(\Delta ABD\right)=\frac{1}{2}ar\left(\Delta ABC\right)$
In $\Delta ABD$, BE is the median.
$\therefore ar\left(\Delta BED\right)=ar\left(\Delta BAE\right)$
$\Rightarrow ar\left(\Delta BED\right)=\frac{1}{2}ar\left(\Delta ABD\right)$
$\Rightarrow ar\left(\Delta BED\right)=\frac{1}{2}\times \frac{1}{2}\times ar\left(\Delta ABC\right)$
$\Rightarrow ar\left(\Delta BED\right)=\frac{1}{4}ar\left(\Delta ABC\right)$