Show that in a right triangle the hypotenuse is the longest side. GIVEN : A right triangle `A B C` in which `/_A B C=90^0dot` TO PROVE : Hypotenuse `A C` is the longest side, i.e. `A C > A B` (ii) `A C > B C`

Explanation

Let us consider a right-angled triangle ABC, right angled at B.
In triangle ABC,
`/_A+/_B+/_C=180^0` (Angle sum property of a triangle)
`/_A +90^0+/_ C = 180^0`
`/_A+/_C = 90^0`
Hence, the other two angles have to be acute (i.e.,less than `90^0`).
Thus, `/_B` is the largest angle in `triangle ABC`. So, `/_B > /_A and /_B> /_C`
Therefore, AC > BC and AC > AB [Using theorem of triangles, in any triangle, the side opposite to the larger (greater) angle is longer.]
Therefore, AC is the largest side in ABC.
However, AC is the hypotenuse of ABC.
Therefore, the hypotenuse is the longest side in a right-angled triangle.