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

$\frac{1}{4}$

When a triangle is formed by joining the midpoints of the sides of the $\Delta$ ABC,
$AD=DB,BF=FC$ and $AE=EC.$
So, we can observe that D divides AB in the ratio $1:1.$

Similarly, E and F also divide AC and BC in the ratio $1:1.$

Hence, $DE\parallel BC,DF\parallel AC$ and $EF\parallel AB$.
So, $\angle ADE=\angle B,\angle AED=\angle C,\angle BDF=\angle A,\angle BFD=\angle C,\angle CFE=\angle CBD$ and $\angle CEF=\angle CAB.$

Therefore, $\Delta ADE\sim \Delta DBF\sim \Delta FED\sim \Delta EFC.$
The scale factor of these triangles is $\frac{AD}{AB}=\frac{1}{2}$ with respect to $\Delta$ ABC.

Hence, the areas of these small triangles is
${\left(\frac{1}{2}\right)}^2\text{Area}\left(\Delta ABC\right)=\frac{1}{4}\text{Area}\left(\Delta ABC\right)$