Question

If $D,E$ and $F$ are respectively the midpoints of sides $AB,BC$ and $CA$ of $\Delta ABC$ then what is the ratio of the areas of $\Delta DEF$ and $\Delta ABC$?

Answer 1

Given

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

To find

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

We know that

The line segment joining the midpoints of any two sides of a triangle is half the third side and parallel to it.

$\therefore FD=\frac{1}{2}BC,ED=\frac{1}{2}AC$and$EF=\frac{1}{2}AB$

In $△ABC$ and $△EFD$, we have

$\frac{AB}{EF}=\frac{BC}{FD}=\frac{AC}{ED}=2...\left(i\right)$

$\Rightarrow △ABC\sim △EFD$[by SSS similarity criterion]

Also, We know that

If two triangles are similar, then the ratio of the area of both triangles is equal to the square of the ratio of their corresponding sides

$\therefore \frac{ar\left(△ABC\right)}{ar\left(△EFD\right)}={\left(\frac{AB}{EF}\right)}^2=4$[from $\left(i\right)$]

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

Hence, the ratio of the areas of $△DEF$ and $△ABC$ is $1:4$