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Question -

Prove that the perimeter of a triangle is greater than the sum of its altitudes



Answer -

Given : In ∆ABC,
AD, BE and CF are altitudes
 
To prove : AB + BC + CA > AD + BC + CF
Proof : We know that side opposite to greater angle is greater.
In ∆ABD, ∠D = 90°
∴ ∠D > ∠B
∴ AB >AD …(i)
Similarly, we can prove that
BC > BE and
CA > CF
Adding we get,
AB + BC + CA > AD + BE + CF

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