If AD is a median of ∆ABC, then the perimeter of ∆ABC cannot be less than or equal to ___________.

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Solution

AD is the median of the ∆ABC.

In ∆ABD,

AB + BD > AD .....(1) (In a triangle, the sum of any two sides is greater than the third side)

In ∆ACD,

CD + CA > AD .....(2) (In a triangle, the sum of any two sides is greater than the third side)

Adding (1) and (2), we get

AB + BD + CD + CA > AD + AD

⇒ AB + BC + CA > 2AD (BC = BD + CD)

⇒ Perimeter of ∆ABC > 2AD

Or the perimeter of ∆ABC cannot be less than or equal to 2AD

If AD is a median of ∆ABC, then the perimeter of ∆ABC cannot be less than or equal to _2AD_.

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