ABCD is a square. Equilateral triangles ACF and ABE are drawn on the the diagona

Question

$A B C D$ is a square. Equilateral triangles $A C F$ and $A B E$ are drawn on the the diagonal $A C$ and side $A B$ respectively. Find area of $△ A C F$ : area of $△ A B E$ .

Options:

A .

\sqrt{2} : 1

B .

2 : 1

C .

4 : 1

D .

8 : 1

Answer: Option B
:
B

Let AB be a units long.
We know that diagonal of a square = $\sqrt{2} \times$ side length
$\Rightarrow A C = \sqrt{2} \times A B$

Now, $△$ ABE and $△$ ACF are equilateral triangles.

Each angle of both $△$ ABE and $△$ ACF is $60^{\circ}$ .

$∴ △ A B E \sim △$ ACF (by AAA similarity criterion)

So, $\frac{a r (A C F)}{a r (A B E)} = \frac{A C^{2}}{A B^{2}} = \frac{2 a^{2}}{a^{2}} = \frac{2}{1}$

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