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ABC is an equilateral triangle. D, E, F are points on BC, CA, AB respectively, such that BD = CE = AF. Then, is it true that $△ DEF$ is an equilateral triangle?

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True

False

answer is A.

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Detailed Solution

Given, ABC is an equilateral triangle. D, E, F are points on BC, CA, AB respectively, such that BD = CE = AF.
Question Image

Since,

△ ABC

is an equilateral triangle, ∠A = ∠B = ∠C = 60°.

\Rightarrow

AB = BC = AC…(1)
AF = BC = AC

\Rightarrow

AF = BD = CE…(2)
Consider (1) - (2).

\Rightarrow

AB - AF = BD - BD = AC - CE

\Rightarrow

BF = CD = AE
In

△ AFE

△ BDF

and

△ CED

,
∠A = ∠B = ∠C = 60°
AF = BD = CE
AE = BF = CD
So,

△ AFE ⩭ △ BDF ⩭ △ CED

.
FE = DF = ED since the three triangles are similar.
Hence,

△ DEF

is an equilateral triangle.
So, the given statement is true and the correct option is 1.

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