Question

Prove that the triangle formed by joining the mid-points of the sides of an equilateral triangle is also equilateral.

Answer 1

Given: Triangle ABC is equilateral.

To Prove: Triangle formed by joining mid-points of sides of ABC is also equilateral.

Let M, N, and P be the mid-points of sides AB, BC, and CA respectively of the equilateral triangle ABC.

By the Mid-Point Theorem:

MN is parallel to AC and MN = `1/2 xx` AC.

NP is parallel to AB and NP = `1/2 xx` AB.

PM is parallel to BC and PM = `1/2 xx` BC.

Since ABC is equilateral, AB = BC = CA.

Therefore,

MN = NP = PM = `1/2 xx` AB (or BC or CA),

and all these sides are equal.

Also, since the sides MN, NP, and PM are parallel respectively to the sides AC, AB, and BC, the angles of triangle MNP correspond to angles of triangle ABC and remain equal.

Hence, triangle MNP is equilateral.

Thus, the triangle formed by joining the mid-points of the sides of an equilateral triangle is also equilateral.