Question

If $\Delta ABC\sim \Delta DFE,\angle A={30}^{\circ },\angle C={50}^{\circ },$ $AB=5cm,AC=8cm,andDF=7.5cm$ ,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) $EF=12cm,\angle D={30}^{\circ }$

Answer 1

(B) $DE=12cm,\angle F={100}^{\circ }$

Given, $\Delta ABC\sim \Delta DFE,then\angle A=\angle D={30}^{\circ },\angle C=\angle E={50}^{\circ }$



In $\Delta ABC$,
$\angle A+\angle B+\angle C={180}^{\circ }$ [Angle sum property]
$\angle B={180}^{\circ }-\angle A-\angle C$
$\angle B={180}^{\circ }-{30}^{\circ }-{50}^{\circ }={100}^{\circ }$
$\therefore$ $\angle B=\angle F={100}^{\circ }$

Also, $AB=5cm,AC=8cm,andDF=7.5cm$
$\therefore \frac{AB}{DF}=\frac{AC}{DE}$
$\Rightarrow \frac{5}{7.5}=\frac{8}{DE}$
$\therefore DE=\frac{8\times 7.5}{5}=12cm$

Hence, $DE=12cm,\angle F={100}^{\circ }$