Question

It is given that ∆ABC ∼ ∆DEF. If ∠A = 30°, ∠C = 50°, AB = 5 cm, AC = 8 cm and DF = 7.5 cm, then which of the following is true?
(a) DE = 12 cm, ∠F = 50°
(b) DE = 12 cm, ∠F = 100°
(c) EF = 12 cm, ∠D = 100°
(d) EF = 12 cm, ∠F = 30°

Answer 1

(b) DE = 12 cm, $\angle$F = 100°

Disclaimer: In the question, it should be ∆ABC ∼ ∆DFE instead of ∆ABC ∼ ∆DEF.

In triangle ABC,
$$\begin{gathered} \angle A+\angle B+\angle C=180° \\ \therefore \angle B=180-30-50=100° \end{gathered}$$

$\because$ ∆ABC ∼ ∆DFE

$$\begin{gathered} \therefore \angle D=\angle A=30°, \\ \angle F=\angle B=100° \\ \mathrm{and}\angle E=\angle C=50° \\ \mathrm{Also}, \\ \frac{AB}{DF}=\frac{AC}{DE} \\ \Rightarrow \frac{5}{7.5}=\frac{8}{DE} \\ \Rightarrow DE=\frac{8\times 7.5}{5}=12\mathrm{cm} \end{gathered}$$