Question

Prove that the hypotenuse is the longest side in a right-angled triangle.

Answer 1

Let us consider a right angled triangle ABC, right angle at B.
In ΔABC
∠A + ∠B +∠C = 180°    ...(angle sum property of a triangle)
∠A + 90° + ∠C = 180°
∠A  +∠C = 90°
Hence, the other two angles have to be acute (i.e. less than 90°).
∴ ∠B is the largest angle in ΔABC.
⇒ ∠B > ∠A and ∠B > ∠C
⇒ AC > BC and AC > AB
[In ant triangle, the side opposite to the larger (greater) angle is longer]
So, Ac is the largest side in ΔABC.
But AC is the hypotenuse of ΔABC. Therefore, hypotenuse is the longest side in a right angled triangle.