The area of an equilateral triangle described on one side of a square is equal to ‘w’ the area of the equilateral triangle described on one of its diagonals. What is the word ‘w’ ?

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Half

Double

Triple

One-Third

answer is A.

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Detailed Solution

Given that, the area of an equilateral triangle described on one side of a square is equal to ‘w’ the area of the equilateral triangle described on one of its diagonals.
Let us represent the given information diagrammatically.
Question Image

Let the side length of the square be ‘a’.
Given that

Δ E D C

and

Δ A C F

are equilateral triangles.
All the equilateral triangles are similar to each other by AAA criteria.
So,

Δ E D C

and

Δ A C F

are similar to each other by AAA similarity.

⇒ Δ E D C ∼ Δ A C F

AC is the diagonal of ABCD, which divides the square into two right angled triangles ADC and ABC.
We know Pythagoras theorem states that in a right-angled triangle,

(h y p o t e n u s e) 2 = (b a s e) 2 + (p e r p e n d i c u l a r) 2

.
In

Δ A D C

, using Pythagoras theorem,

A C 2 = A D 2 + D C 2

Now in square ABCD, AD = DC=a,

⇒ A C 2 = a 2 + a 2 ⇒ A C 2 = 2 a 2

⇒ A C = a 2

We know that if two triangles are similar, then ratio of their areas is in proportion with the ratio of squares of their corresponding sides.
Using the properties of similar triangles,

a r (Δ D E C) a r (Δ A C F) = D C 2 A C 2

Putting in the values, we get

⇒ a r (Δ D E C) a r (Δ A C F) = a 2 (2 a) 2 ⇒ a r (Δ D E C) a r (Δ A C F) = a 2 2 a 2

⇒ a r (Δ D E C) a r (Δ A C F) = 12

⇒ a r (Δ D E C) = 12 a r (Δ A C F)

Area of the equilateral triangle on the side of the square is half of the area of the equilateral triangle on the diagonal of the square.
Therefore the correct option is 1.

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