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Question
Show that in a right angled triangle, the hypotenuse is the longest side.
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Solution
Let us consider a right-angled triangle ABC, right-angled at B.
In ΔABC,
∠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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