Question

o
1 L,Mand N are the mid point of the sides QR, RP and PQ of
an equilateral triangle PQR prove than triangle LMN is
also an equilateral triangle

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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.

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

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

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

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

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

∴ LN = $\frac{1}{2}$ BC, LM= $\frac{1}{2}$, MN= $\frac{1}{2}$ AB

Now,△ABC is an equilateral triangle,

∴AB=BC=CA

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

⇒MN=LN=LM

Hence △LMN is an equilateral triangle.

Hope you understand