Question

D, E, F are the midpoints of the sides BC, CA and AB respectively of $△$ ABC. Then $△$ DEF is congruent to triangle –

• A. ABC
• B. AEF
• C. BFD, CDE
• D. AFE, BFD, CDE

Answer 1

Answer: (B), (C), (D)

The correct options are
B

AEF

C

BFD, CDE

D

AFE, BFD, CDE

It is easy to verify that $△$ DEF is congruent to each one of the triangles $△$ AEF, $△$ BFD & $△$ CDE.

Consider $\Delta AFE\angle AFE=\angle Band\angle AEF=\angle C$

Since FE is parallel to BC.

Similarly in $\Delta BFD\angle BFD=\angle Aand\angle BDF=\angle C$ and

$\Delta DEC\angle DEC=\angle Aand\angle EDC=\angle B$

We know $\angle A+\angle B+\angle C={180}^{\circ }$

Now, consider $\Delta DEF\angle FED={180}^{\circ }-\angle DBF-\angle EDC=180-\angle C-\angle B=\angle A$
Similarly, $\angle DFE=\angle C$ and $\angle FED=\angle B$.
By AA similarity
So, $\Delta DEF$~$\Delta AFE$~$\Delta BFD$~$\Delta CDE$