Question

In $\Delta$ABC, if a triangle is formed by joining the midpoints of the sides, the area of the triangle is___ times the area of $\Delta$ ABC.

• A. 2
• B. $\frac{1}{3}$
• C. $\frac{1}{4}$
• D. 4

Answer 1

The correct option is C

$\frac{1}{4}$

Here, D, E and F are midpoints of sides AB, AC and BC respectively.

$\Rightarrow$ AD = DB, BF = FC and AE = EC.

We know that the line joining the midpoints of two sides of a triangle is parallel to the third side and is half of the third side (midpoint theorem).

So, BC = 2DE
AC = 2DF
AB = 2EF

In $△ABC$ and $△DEF$,

$\frac{AB}{EF}=\frac{AC}{DF}=\frac{BC}{DE}=\frac{2}{1}$

$\therefore △ABC\sim △FED$ (by SSS similarity criterion)

Now, $\frac{ar\left(△ABC\right)}{ar\left(△FED\right)}=\frac{A{B}^2}{E{F}^2}=\frac{2^2}{1^2}$

$\Rightarrow ar\left(△ABC\right)=4\times ar\left(△FED\right)$

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