question_answer

It is given that \[\Delta ABC\tilde{\ }\Delta DFE,\] \[\angle A=30{}^\circ ,\] \[\angle C=50{}^\circ ,\] \[AB=5cm,\]and \[DF=7.5cm\]. Then, which of the following is true?

A) \[DE=12cm,\,\,\,\angle F=50{}^\circ \]

B) \[DE=12cm,\,\,\,\angle F=100{}^\circ \]

C) \[EF=12cm,\,\,\,\angle D=100{}^\circ \]

D) \[EF=12cm,\,\,\,\angle D=30{}^\circ \]

Correct Answer: B

Solution :

[b] Given,   \[\Delta ABC-\Delta DEF,\]

then      \[\angle A=\angle D=30{}^\circ \]

\[\angle C=\angle E=50{}^\circ \]

\[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\angle B=\angle F=180{}^\circ -(30{}^\circ +50{}^\circ )=100{}^\circ \]

Also, \[AB=5\,cm,\] \[AC=8\,cm\] and \[DF=7.5\,cm\]

\[\therefore \,\,\,\,\,\,\,\,\frac{AB}{DF}=\frac{AC}{DE}\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\,\,\,\frac{5}{7.5}=\frac{8}{DE}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,DE=\frac{8\times 7.5}{5}=12\,cm\]

Hence. \[DE=12cm,\,\,\,\angle F=100{}^\circ \]