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Question
In the given figure, ΔABC is an equilateral traingle. Points F, D and E are midpoints of side AB, side BC, side AC respectively. Show that ΔFED is an equilateral traingle.
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Solution
Given: ∆ABC is an equilateral triangle and D, E and F are mid-points of BC, AC and AB respectively.
To prove: ∆FED is an equilateral triangle.
Proof:
In ΔABC,
Points F and E are the midpoints of sides AB and AC respectively. ...(Given)
∴ FE = `1/2` BC ...(From midpoint theorem) ...(i)
In ΔABC,
Points D and E are the midpoints of sides BC and AC respectively. ...(Given)
∴ DE = `1/2` AB ...(From midpoint theorem) ...(ii)
In ΔABC,
Points D and F are the midpoints of sides BC and AB respectively. ...(Given)
∴ DF = `1/2` AC ...(From midpoint theorem) ...(iii)
Now, ΔABC is an equilateral triangle.
∴ BC = AB = AC ...(Sides of equilateral triangle)
∴ `1/2` BC = `1/2` AB = `1/2` AC ...(Multiplying both sides by `1 /2`)
∴ FE = DE = DF ...[From (i), (ii) and (iii)]
∴ ΔFED is an equilateral triangle.
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