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Grade: 10
        Prove that the area of equilateral triangle described on side of a square is half the area of equilateral triangle described on its diagonals
9 months ago

Answers : (1)

Arun
13923 Points
							
Dear Vaibhav
 
I am not attaching any figure.but you can assume related to my answer.
 
Here ABCD is a square, AEB is an equilateral triangle described on the side of the square and DBF is an equilateral triangle described on diagonal BD of  the square.
We have to Prove: Ar(ΔDBF) / Ar(ΔAEB) = 2 / 1 
Proof:  If two equilateral triangles are similar then all angles are = 60 degrees.

Therefore, by AAA similarity criterion , △DBF ~

 △AEB

Ar(ΔDBF) / Ar(ΔAEB) = DB2 /AB2 -----------1

We know that the ratio of the areas of two similar triangles is equal to
the square of the ratio of their corresponding sides i .e.
But, we have DB = √2AB     {But diagonal of square is √2 times of its side} -----2
Substitute equation 2 in equation 1, we get
Ar(ΔDBF) / Ar(ΔAEB) = (√2AB )2 / AB2   = 2 AB2 / AB2 = 2

∴ Area of equilateral triangle described on one side os square is equal to half the area of the equilateral triangle described on one of its diagonals.   
 
 
Regards
Arun (askIITians forum expert)
9 months ago
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