Question

$ABC$ is an equilateral triangle and $L,M$ and $N$ are the mid points of the sides $AB,BC$ and $CA$, respectively then $△LMN$ is an equilateral triangle

• A. True
• B. False

Answer 1

The correct option is A True

Given:$△ABC$ is an equilateral triangle and $L,M,N$ are the mid-points of the sides $AB,BC$ and $CA$ respectively.

Since $L$ and $N$ are the mid-points of $AB$ and $AC$ respectively.

$\therefore LN=\frac{1}{2}BC$ (by mid-point theorem)

Since $L$ and $M$ are the mid-points of $AB$ and $BC$ respectively.

$\therefore LM=\frac{1}{2}AC$(by mid-point theorem)

Since $M$ and $N$ are the mid-points of $BC$ and $AC$ respectively.

$\therefore MN=\frac{1}{2}AB$(by mid-point theorem)

$\therefore LN=\frac{1}{2}BC,LM=\frac{1}{2}AC,MN=\frac{1}{2}AB$ .........$\left(1\right)$

Now,$△ABC$ is an equilateral triangle,

$\therefore AB=BC=CA$

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

$\Rightarrow MN=LN=LM$

Hence $△LMN$ is an equilateral triangle.

Hence the given statement is true.