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Question:

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

Solution:

Consider ΔPQR which is right angled at Q. By angle sum property of a triangle, ∠PQR + ∠PRQ + ∠QPR = 180° ⇒ 90° + ∠R + ∠P = 180° ⇒ ∠R + ∠P = 90° ⇒ ∠R and ∠P are acute angles ⇒ ∠R < 90° and ∠P < 90° ⇒ ∠R < ∠Q and ∠P < ∠Q ⇒ PR > PQ and PR > QR (Side opposite to greater angle is greater) So, the hypotenuse is the longest side in a triangle.