Question

If the altitudes from two vertices of a triangle to the opposite sides are equal, then the triangle is
(a) equilateral
(b) isosceles
(c) scalene
(d) right-angled

Answer 1

(b) isosceles


In $∆ABC,\mathrm{BL}\parallel \mathrm{AC}.$
$\mathrm{CM}\parallel \mathrm{AB}\mathrm{such}\mathrm{that}\mathrm{BL}=\mathrm{CM}.$

To prove: AB = AC

$$\begin{gathered} \mathrm{In}△\mathrm{ABL}\mathrm{and}△\mathrm{ACM}, \\ \mathrm{BL}=\mathrm{CM}\left(\mathrm{Given}\right) \\ \angle \mathrm{BAL}=\angle \mathrm{CAM}(\mathrm{Common}\mathrm{angle}) \\ \angle \mathrm{ALB}=\angle \mathrm{AMC}(\mathrm{Each}90°) \\ △\mathrm{ABL}\cong △\mathrm{ACM}(\mathrm{AAS}\mathrm{criterion}) \\ \therefore \mathrm{AB}=\mathrm{AC}\left(\mathrm{CPCT}\right) \end{gathered}$$