Question

A. If D, E, F are respectively the mid-points of the sides AB, BC and CA of an equilateral triangle
ABC, prove that ADEF is also an equilateral triangle.
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In the fig, D, E and F are, respectively the mid-points of sides BC, CA and AB of an equilateral triangle ABC.

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asked Dec 23, 2017 in Class IX Maths by ashu Premium (930 points)

In the fig, D, E and F are, respectively the mid-points of sides BC, CA and AB of an equilateral triangle ABC. Prove that DEF is also an equilateral triangle.

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answered Dec 23, 2017 by navnit40 (-4,939 points)

Since line segment joining the mid-points of two sides of a triangle is half of the third side. Therefore, D and E are mid-points of BC and AC respectively.

⇒ DE = 1 / 2 AB --- (i)

E and F are the mid - points of AC and AB respectively .

∴ EF = 1 / 2 BC --- (ii)

F and D are the mid - points of AB and BC respectively .

∴ FD = 1 / 2 AC --- (iii)

Now, △ABC is an equilateral triangle .

⇒ AB = BC = CA

⇒ 1 / 2 AB = 1 / 2 BC = 1 / 2 CA

⇒ DE = EF = FD [using (i) , (ii) , (iii) ]

Hence, DEF is an equilateral triangle

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