Question

In a right-angled triangle, prove that the hypotenuse is the longest side.

Answer 1

$$\begin{gathered} \mathrm{Let}∆ABC\mathrm{be}\mathrm{a}\mathrm{right}-\mathrm{angle}d\mathrm{triangle},\mathrm{in}\mathrm{which}\angle C\mathrm{is}\text{the}\mathrm{right}\mathrm{angle}. \\ \mathrm{In}∆ABC, \\ \angle ACB=90° \\ \angle ACB+\angle CBA+\angle BAC=180°\left(\mathrm{angle}\mathrm{sum}\mathrm{property}\right) \\ \Rightarrow 90°+\angle CBA+\angle BAC=180° \\ \Rightarrow \angle CBA+\angle BAC=180°-90° \\ \Rightarrow \angle CBA+\angle BAC=90° \\ \text{So,}\angle CBA\mathrm{and}\angle BAC\mathrm{are}\mathrm{acute}\mathrm{angles}. \\ \Rightarrow \angle CBA<90°\mathrm{and}\angle BAC<90° \\ \Rightarrow \angle CBA<\angle ACB\mathrm{and}\angle BAC<\angle ACB \\ \Rightarrow AB\hspace{0.17em}>AC\mathrm{and}AB>BC\left(\mathrm{side}\mathrm{opposite}\mathrm{to}\mathrm{greater}\mathrm{angle}\mathrm{is}\mathrm{greater}\right) \\ \mathrm{Therefore},\mathrm{hypotenuse}AB\mathrm{is}\text{the}\mathrm{longest}\mathrm{side}. \\ \mathrm{Hence},\mathrm{proved}. \end{gathered}$$