Question

ABC is an equilateral triangle and L,M and N are the mid points of the sides AB,BC and CA, respectively.Prove that ∆LMN is an equilateral triangle

Answer 1

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Answer 2

ΔLMN is an equilateral traingle.

GIVEN

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

TO PROVE

∆LMN is an equilateral triangle.

SOLUTION

We can simply solve the above problem as follows;

It is given that ΔABC is an equilateral triangle.

Let,

AB = BC = CA = x (Sides of equilateral triangle are equal)

L is the midpoint of Side AB

N is the midpoint of side BC

Applying mid-point theorem

LN || BC

And,

LN = 1/2BC = 1/2x (I)

Similarly,

M is the midpoint of BC

N is the midpoint of AC

Therefore,

NM || AB

And,

NM = 1/2AB = 1/2x (II)

L is the mid-point of AB

M is the mid-point of BC

So,

LM || AC

And,

LM = 1/2AC = 1/2x (III)

From (I), (II), and (III)

LN = LM = MN = 1/2x

Since all the sides of the triangle ΔLMN are equal,

ΔLMN is an equilateral traingle.

ΔLMN is an equilateral traingle. Hence, Proved.

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