Choose the correct answer from the given four options:
It is given that $ \triangle \mathrm{ABC} \sim \triangle \mathrm{DFE}, \angle \mathrm{A}=30^{\circ}, \angle \mathrm{C}=50^{\circ}, \mathrm{AB}=5 \mathrm{~cm}, \mathrm{AC}=8 \mathrm{~cm} $ and $ D F=7.5 \mathrm{~cm} $. Then, the following is true:
(A) $ \mathrm{DE}=12 \mathrm{~cm}, \angle \mathrm{F}=50^{\circ} $
(B) $ \mathrm{DE}=12 \mathrm{~cm}, \angle \mathrm{F}=100^{\circ} $
(C) $ \mathrm{EF}=12 \mathrm{~cm}, \angle \mathrm{D}=100^{\circ} $
(D) $ \mathrm{EF}=12 \mathrm{~cm}, \angle \mathrm{D}=30^{\circ} $
Given:
\( \triangle \mathrm{ABC} \sim \triangle \mathrm{DFE}, \angle \mathrm{A}=30^{\circ}, \angle \mathrm{C}=50^{\circ}, \mathrm{AB}=5 \mathrm{~cm}, \mathrm{AC}=8 \mathrm{~cm} \) and \( D F=7.5 \mathrm{~cm} \).
To do:
We have to choose the correct answer.
Solution:
From the figure,
$\angle B=\angle F$
$=180^{\circ}-(30^{\circ}+50^{\circ})$
$=100^{\circ}$
$A B=5 \mathrm{~cm}, A C=8 \mathrm{~cm}$ and $DF=7.5 \mathrm{~cm}$
Therefore,
$\frac{A B}{D F}=\frac{A C}{D E}$
$\frac{5}{7.5}=\frac{8}{D E}$
$D E=\frac{8 \times 7.5}{5}$
$=12 \mathrm{~cm}$
Therefore,
$D E=12 \mathrm{~cm}, \angle F=100^{\circ}$
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