Question

In $\Delta ABC$, $D,E,F$ are respectively the midpoints of the sides $AB,BC$ and $AC$.

Find $\frac{\text{Area of}\Delta DEF}{\text{Area of}\Delta ABC}$.

• A. $2:1$
• B. $3:4$
• C. $2:3$
• D. $1:4$

Answer 1

The correct option is D $1:4$

Given in $△ABC$ $D,E,F$ are the mid points of sides $AB,BC$ and $CA$ respectively

Now using mid point theorem line segment joining the mid points of two sides is parallel to third side and also half of it.

$\therefore DF=\frac{1}{2}BC$

$\Rightarrow \frac{DF}{BC}=\frac{1}{2}$ ....(i)

Similarly $\frac{DE}{AC}=\frac{1}{2}$ ....(ii)

$\frac{EF}{AB}=\frac{1}{2}$ ....(iii)

Using (i), (ii) and (iii), we have

$\frac{DF}{BC}=\frac{DE}{AC}=\frac{EF}{AB}=\frac{1}{2}\therefore △ABC\sim △DEF$

Now if triangles are similar then ratio of their areas is equal to ratio of square of their corresponding sides.

$\frac{ar\left(△DEF\right)}{ar\left(△ABC\right)}={\left(\frac{1}{2}\right)}^2=\frac{1}{4}$