Question

In the given figure, D is the midpoint of BC, DE ⊥ AB and DF ⊥ AC such that DE = DF. Then, which of the following is true?
(a) AB = AC
(b) AC = BC
(c) AB = BC
(d) None of these

Answer 1

(a) AB = AC

$$\begin{gathered} \mathrm{In}△\mathrm{BED}\mathrm{and}△\mathrm{CFD},\mathrm{we}\mathrm{get}: \\ \mathrm{DE}=\mathrm{DF}\left(\mathrm{Given}\right) \\ \mathrm{BD}=\mathrm{CD}\left(\mathrm{Given}\right) \\ \angle \mathrm{E}=\angle \mathrm{F}=90°\left(\mathrm{Given}\right) \\ \therefore △\mathrm{BED}\cong △\mathrm{CFD}(\mathrm{RHS}\mathrm{criterion}) \\ \Rightarrow \angle \mathrm{B}=\angle \mathrm{C} \\ \Rightarrow \mathrm{AC}=\mathrm{AB} \end{gathered}$$