Question

given Triangle LMN is an equilateral triangle and LA is perpendicular toMN prove that LA=√3MA​

Answer 1

Answer:

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.

∴LN=

2

1

BC (by mid-point theorem)

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

∴LM=

2

1

AC(by mid-point theorem)

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

∴MN=

2

1

AB(by mid-point theorem)

∴LN=

2

1

BC,LM=

2

1

AC,MN=

2

1

AB .........(1)

Now,△ABC is an equilateral triangle,

∴AB=BC=CA

⇒

2

1

AB=

2

1

BC=

2

1

CA

⇒MN=LN=LM

Hence △LMN is an equilateral triangle.

Hence the given statement is true.

solution