Question

m is a point on side Bc of a ∆ABC such that AM is the bisector of angle BAC .is it true to say that perimeter of the triangle is greater than 2AM. Give reason.

Answer 1

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Sai Sandhya asked in Math

M is a point on side BC of a triangle ABC such that AM is the bisector of angle BAC.
Is it true to say that perimeter of the triangle is greater than 2AM?Give reason for your
answer.

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Ankush Jain answered this
21515 helpful votes in Math, Class XII-Science



Given: In triangle ABC, AM is the bisector of angle BAC.

To prove: AB + BC + AC > 2 AM

Proof: 

In triangle ABM, we have

AB + BM > AM  [Sum of two sides of any triangle is always greater than its third side]  .... (1)

Similarly, in triangle AMC , we have

AC + CM > AM  [Sum of two sides of any triangle is always greater than its third side]  .... (2)

On adding (1) and (2), we get

AB + BM + AC +CM > AM + AM 

⇒ AB + AC  + (BM + CM) > AM + AM

⇒ AB + AC  + BC > 2AM  [Hence proved]

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