Question

In $\Delta ABC$, D, E and F are midpoints of $\overline{AB},\overline{BC}$ and $\overline{AC}$ respectively. If $A\left(\Delta ABC\right)=40$ then $A\left(\Delta DEF\right)=$ ..................

• A. $10$
• B. $\frac{40}{3}$
• C. $20$
• D. $5$

Answer 1

The correct option is A $10$

$D,E$ and $F$ are the mid-points of the sides $\overline{AB},\overline{BC}$ and $\overline{CA}$ of $\Delta ABC$.
$FD\parallel BC$
$FD=\frac{1}{2}BC\Rightarrow \frac{FD}{BC}=\frac{1}{2}$
$DE\parallel AC$
$DE=\frac{1}{2}AC\Rightarrow \frac{DE}{AC}=\frac{1}{2}$
$EF\parallel AB$
$EF=\frac{1}{2}AB\Rightarrow \frac{EF}{AB}=\frac{1}{2}$
$\therefore ABC\sim EDF$ ....... [By SSS test of similarity]
The ratios of areas of similar triangles is equal to the squares of the ratio of their corresponding sides.
$\Rightarrow \frac{A\left(△DEF\right)}{A\left(△ABC\right)}={\left(\frac{EF}{AB}\right)}^2={\left(\frac{1}{2}\right)}^2$
$\Rightarrow \frac{A\left(△DEF\right)}{40}=\frac{1}{4}$
$\Rightarrow A\left(△DEF\right)=10$
$\therefore$ Area of $△DEF$ is 10 sq. units.