Question

Question 8
D, E and F are the mid – points of the sides BC, CA and AB, respectively of an equilateral $\Delta ABC$, show that $\Delta DEF$ is also an equilateral triangle.

Answer 1

Given in equilateral $\Delta ABC.D,E$, and F are the mid-points of sides BC, CA and AB. Respectively.

To show $\Delta DEF$ is an equilateral triangle.

Proof Since in $\Delta ABC$, E and F are the mid-points of AC and AB respectively, then EF ∥ BC and

$EF=\frac{1}{2}BC$ …(i)

Similarly DF ∥ AC, DE ∥ AB

And $DE=\frac{1}{2}ABandFD=\frac{1}{2}AC$ [ By mid – point theorem] ….(ii)

Since $\Delta ABC$ is an equilateral triangle

AB=BC=CA [dividing by 2 ]

$\Rightarrow \frac{1}{2}AB=\frac{1}{2}BC=\frac{1}{2}CA$

DE=EF=FD [from eqs. (i) and (ii)]

Thus, all sides of $\Delta DEF$ are equal.

Hence, $\Delta DEF$ is an equilateral triangle. Hence proved.