Question

prove that the area of an equilateral triangle described on one side of a square is equal to half the area of equilateral triangle described in on one of its diagonals . ​

Answer 1

Answer:

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Answer 2

Given: A square ABCD in which BD is the diagonal and two Δs namely ΔFCD and ΔEBD are described.

To prove: Ar(ΔFCD) = 1/2 Ar(ΔEBD)

Proof: We know that diagonal of a square is

$\sqrt{2} \times side$

and, area of an equilateral triangle is

$\frac{ \sqrt{3} }{4} \times {(side \: of \: triangle)}^{2}$

therefore, Ar(ΔEBD) =

$\frac{ \sqrt{3} }{4} \times { (\sqrt{2} )}^{2} = \frac{ \sqrt{3} }{4} \times 2$

and Ar(ΔFDC) =

$\frac{ \sqrt{3} }{4} \times 1$

therefore, Ar(ΔFCD) = 1/2 Ar(ΔEBD)

Proved....

Hope it would help.