Question

If fig.D,E and F are respectively the midpoints of sides BC, CA and AB of an equilateral triangles ABC. Prove that DEF is also an equilateral triangle​

Answer 1

The above Question.can be proved with the help of mid-point theorem.

Given : in ∆ABC

AB = BC = CA

also D, E & F are the mid points.

To prove: DEF is an equilateral triangle.

Proof: AB = BC = CA ....... (given)

(multiply the equation with 1/2)

so, 1/2 AB = 1/2 BC = 1/2 CD

or, DE = EF = FD ....(1)

So, in ∆DEF

DE = EF = FD ....... (from eq.1)

Hence Proved

HOPE IT WAS HELPFUL TO YOU !

Answer 2

${\huge{\underline{\underline{\sf{\red{Solution:-}}}}}}$

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Since the 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.

$⇒⠀⠀⠀⠀DE= \frac{1}{2} AB$

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

$∴⠀⠀⠀EF= \frac{1}{2} BC$

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

$⇒⠀⠀⠀⠀FD= \frac{1}{2} AC$

∵⠀∆ABC is an equilateral triangle

∴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀AB = BC = CA

$⇒⠀⠀⠀⠀ \frac{1}{2}AB = \frac{1}{2} BC= \frac{1}{2}C A$

⇒⠀⠀⠀DE = EF = FD ⠀⠀⠀⠀⠀[Using (1),(2), (3)]

Hence,

∆ DEF is an equilateral triangle.⠀⠀⠀ Proved.

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