$A B C D$ 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 $△ A B E$ .

$\sqrt{2} : 1$

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$2 : 1$

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$4 : 1$

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$8 : 1$

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Solution

The correct option is B
$2 : 1$

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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