Question

D, E and F are respectively mid points of sides $\overline{AB},\overline{BC}$ and $\overline{CA}$ of $\Delta ABC$. Which of the following correspondence between $\Delta ABC$ and $\Delta DEF$ of similarity.

• A. $ABC\sim EFD$
• B. $ABC\sim FED$
• C. $ABC\sim DEF$
• D. $ABC\sim EDF$

Answer 1

The correct option is A $ABC\sim EFD$

D, E and F are the mid-points of the sides AB, BC and CA of $\Delta$ ABC.
FD || BC
$FD=\frac{1}{2}BC\Rightarrow \frac{FD}{BC}=\frac{1}{2}$
DE || AC
$DE=\frac{1}{2}AC\Rightarrow \frac{DE}{AC}=\frac{1}{2}$
EF || AB
$EF=\frac{1}{2}AB\Rightarrow \frac{EF}{AB}=\frac{1}{2}$

Then, quadrilaterals $ADEF,BEFD,CFDE$ will be $||gm$
$\Rightarrow \angle A=\angle Eand\angle B=\angle Fand\angle C=\angle D$
$\therefore ABC\sim EFD$