Question : ABC is an equilateral triangle points D, E and F are taken in sides AB, BC and CA respectively so that, AD = BE = CF. Then DE, EF, and FD enclose a triangle which is:
Option 1: equilateral
Option 2: isosceles
Option 3: right angled
Option 4: none
Correct Answer: equilateral
Solution :
Given: ABC is an equilateral triangle points D, E and F are taken in sides AB, BC and CA respectively so that, AD = BE = CF.
Now, AB = BC = CA and AD = BE = CF
So, AD = BD = BE = EC = CF = AF
⇒D, E, and F are midpoints of the sides AB, BC and AC, respectively.
Thus, DF || BC and DF = $\frac{1}{2}$BC
Similarly DE = $\frac{1}{2}$AC and EF = $\frac{1}{2}$AB
So, DE = EF = DF
$\therefore \triangle$DEF is an equilateral triangle.
Hence, the correct answer is equilateral.
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