Question

D,E,F are midpoints of sides BC, CA and AB of $\Delta ABC$. If perimeter of $\Delta ABC$ is 12.8 cm, then perimeter of $\Delta DEF$ is :

• A. $17cm$
• B. $38.4cm$
• C. $25.6cm$
• D. $6.4cm$

Answer 1

The correct option is D $6.4cm$

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.

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

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

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

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

Using $\left(i\right),\left(ii\right)$ and $\left(iii\right)$

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

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

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

$\frac{Perimeter\left(△DEF\right)}{Perimeeter\left(△ABC\right)}=\frac{1}{2}$

$\Rightarrow$ Perimeter of $△ABC=\frac{1}{2}\times 12.8=6.4$ cm