4 Mark Question

Question

In the adjoining figure, ∆ABC is an equilateral triangle. Points F, D and E are midpoints of side AB, side BC, side AC respectively. Show that ∆FED is an equilateral triangle.

Given: ∆ABC is an equilateral triangle.
Points F, D and E are midpoints of side AB, side BC, side AC respectively.
To prove: ∆FED is an equilateral triangle.

Accepted Answer

Proof:
∆ABC is an equilateral triangle. [Given]
∴ AB = BC = AC ….(i) [Sides of an equilateral triangle]
Points F, D and E are midpoints of side AB and BC respectively.
$
\therefore FD =\frac{1}{2} AC \ldots \text {...(ii) [Midpoint theorem] }
$
Points $D$ and $E$ are the midpoints of sides $B C$ and $A C$ respectively.
$
\therefore DE =\frac{1}{2} AB \text {....(iii) [Midpoint theorem] }
$
Points $F$ and $E$ are the midpoints of sides $A B$ and $A C$ respectively.
$
\therefore F E=\frac{1}{2} BC
$
$\therefore FD = DE = FE [$ [From (i), (ii), (iii) and (iv) ]
$\therefore \triangle FED$ is an equilateral triangle.

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