Raji
2

As we know, area of an equilateral Δ=√3/4×(side)²

∆=√3/4×(side)²

Let us assume

side of square = a

eq. ∆ drawn on side of sq. = ∆¹

eq. ∆ drawn on diagonal = ∆²

As we know, the diagonal of the square = √2×a

( by applying Pythagoras Theorem)

So, area of ∆¹ = √3/4×

area of ∆² √3/4×(√2 )² =2[√3/4×]


As we can clearly see


area of ∆² =2×∆¹

Hence Proved.


Let us use an example as shown below:

Here ABCD is a square, AEB is an equilateral triangle (∆¹) described on the side of the square and DBF is an equilateral triangle (∆²) described on diagonal BD of the square.


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  --------------------(i)


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} -----(ii).


Substitute equation (ii) in equation (i), we get


Ar(ΔDBF) / Ar(ΔAEB) = (√2AB )2 / AB2  = 2 AB/ AB= 2


∴ Area of equilateral triangle described on one side is square is equal to half the area of the equilateral triangle described on one of its diagonals.

Krishna
1

STEP 1:  Know about the equilateral triangle and it's properties


(https://brilliant.org/wiki/properties-of-equilateral-triangles/)


STEP 2: Use the given data in the question to make a figure.


STEP 3: Prove two equilateral triangles are similar [one Δ is on the side of the square and another one is on the diagonal]

              (To prove this recall the criteria for similarity of triangles 

1) AAA or AA 2) SSS 3) SAS. )


STEP 4: Conclude that the ratio of the areas of two similar triangles is equal to

the square of the ratio of their corresponding sides. If the equilateral triangles are similar.

\frac{Area(Δ on diagonal)}{Area( Δ on side)} = \frac{diagonal^2}{ side^2}


STEP 5: Use the Pythagoras theorem to calculate the diagonal of the square.

  (The diagonal of square is √2 times of its side)


STEP 6: Substitute step 5 value in step 4 and simplify to prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals