Question

Question 1
Show that in a right angled triangle, the hypotenuse is the longest side.

Answer 1

Let us consider a right-angled triangle ABC, right-angled at B.
In $\Delta$ABC,
$\angle A+\angle B+\angle C={180}^{\circ }$ (Angle sum property of a triangle)
$\angle A+{90}^{\circ }+\angle C={180}^{\circ }$
$\angle A+\angle C={90}^{\circ }$
Hence, the other two angles have to be acute (i.e., less than ${90}^{\circ }$
$\therefore \angle B$ is the largest angle in $\Delta$ABC.
$\Rightarrow \angle B$ > $\angle A$ and $\angle B$ > $\angle C$
$\Rightarrow$ AC > BC and AC > AB
[In any triangle, the side opposite to the larger (greater) angle is longer.]
Therefore, AC is the largest side in $\Delta$ABC.
However, AC is the hypotenuse of $\Delta$ABC. Therefore, hypotenuse is the longest side in a right-angled triangle.