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In the figure...

Question

In the figure below, $∠ A B C = ∠ D F E, ∠ B A C = ∠ F D E$ , $D$ and $F$ are on $A B, A D = F B$ and the distances are shown in the figure itself in centimeters. Calculate the length of $A D$ , in centimeters.

5

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Solution

The correct option is E $2$
In $△ A B C$ and $△ D F E,$
$∠ A B C = ∠ D F E$ [Given]
$∠ B A C = ∠ F D E$ [Given]

Hence, by $A A A$ rule of similarity,
$△ A B C \sim △ D F E$

As, the two triangles are similar their sides will be proportional as well,
$∴ \frac{A B}{D F} = \frac{B C}{F E} = \frac{A C}{D E} \dots (i)$

Let $A D = B F = x$
$A B = A D + D F + B F$
$A B = 6 + 2 x$

From equation $(i),$
$\begin{matrix} \frac{A B}{D F} & = \frac{A C}{D E} \frac{6 + 2 x}{6} & = \frac{20}{12} 72 + 24 x & = 120 24 x & = 48 x & = 2 \end{matrix}$
Hence,
$A D = B F = 2 c m .$

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