Maths 1 Marks Question

To show $\Delta\text{DEF}$ is an equilateral triangle.

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

and $\text{EF}=\frac{1}{2}\text{BC}\ ...(\text{i})$

DF || AC, DE || AB

$\text{DE}=\frac{1}{2}\text{AB}$ and $\text{FD}=\frac{1}{2}\text{AC}$ [by mid-point theorem] …(ii)

since $\Delta\text{ABC}$ is an equilateral triangle

AB = BC = CA

$\Rightarrow\ \frac{1}{2}=\frac{1}{2}\text{BC}=\frac{1}{2}\text{CA}$ [dividing by 2]

$\Rightarrow\ \therefore\ \text{DE}=\text{EF}=\text{FD}$ [from Eqs. (i) and (ii)]

Thus, all sides of ADEF are equal.

Hence, $\Delta\text{DEF}$ is an equilateral triangle.

Hence proved.