Question

Show that in a right angled triangle, the hypotenuse is the longest side.

Answer 1

$\because Sumofanglesoftriangle={180}^{\circ }$

$\Rightarrow \alpha +\beta ={90}^{\circ }$

$\Rightarrow \alpha <{90}^{\circ },\beta <{90}^{\circ }$

$sin\left(\alpha \right)=\frac{AB}{AC}$

$\Rightarrow AC=\frac{AB}{sin\alpha }$

$sin\alpha ϵ(0,1)$

$\therefore \frac{1}{sin\alpha }>1$

$\Rightarrow AC>AB-\left(1\right)$

$sin\beta =\frac{BC}{AC}$

$\Rightarrow AC=\frac{BC}{sin\beta }$

$sin\beta \in (0,1)$

$\Rightarrow \frac{1}{sin\beta }>1$

$\Rightarrow AC>BC-\left(2\right)$

From(1)\& (2),AC is the longest side.