Show that in a right angled triangle, the hypotenuse is the longest side.
Let us consider a right-angled triangle ABC, right-angled at B.
∠A + ∠B + ∠C = 180° (Angle sum property of a triangle)
∠A + 90º + ∠C = 180°
∠A + ∠C = 90°
Hence, the other two angles have to be acute (i.e., less than 90º).
∴ ∠B is the largest angle in ΔABC.
⇒ ∠B > ∠A and ∠B > ∠C
⇒ 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 ΔABC.
However, AC is the hypotenuse of ΔABC. Therefore, hypotenuse is the longest side in a right-angled triangle.
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- Inequalities in a Triangle