Question

In a $△ABC$, the midpoints of sides $BC,CA$ and $AB$ are $D,E$ and $F$ respectively. Find ratio of $ar\left(△DEF\right)$ to $ar\left(△ABC\right)$

Answer 1

Solution:

Given that:

$D,E$ and $F$ are mid-points of $BC,CA$ and $AB.$

To Find:

$\frac{ar\left(△DEF\right)}{ar\left(△ABC\right)}=?$

Solution:

Since, $D,E$ and $F$ are mid-points of $BC,CA$ and $AB.$

or, $BD=CD,AE=CE$ and $AF=BF$

$E$ and $F$ are mid-points of $CA$ and $AB.$

$FEDB$ is a llgm. (By midpoint theorem.)

$\therefore ar\left(△DEF\right)=ar\left(△BFD\right)$

Similarly,

$ar\left(△DEF\right)=ar\left(△DEC\right)$

$ar\left(△DEC\right)=ar\left(△AFE\right)$

$\therefore ar\left(△DEF\right)=ar\left(△BFD\right)=ar\left(△DEC\right)=ar\left(△AFE\right)$

Now,

$ar\left(△ABC\right)=ar\left(△DEF\right)+ar\left(△BFD\right)+ar\left(△DEC\right)+ar\left(△AFE\right)$

or, $ar\left(△ABC\right)=4ar\left(△DEF\right)$

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