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Question
M is a point on side BC of a triangle ABC such that AM is the bisector of ∠BAC. Is it true to say that perimeter of the triangle is greater than 2 AM? Give reason for your answer.
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Solution
Yes, In ΔABC, M is a point of side BC such that AM is the bisector of ∠BAC.
In ΔABM, AB + BM > AM ...(i) [Sum of two sides of a triangle is greater than the third side]
In ΔACM, AC + CM > AM ...(ii) [Sum of two sides of a triangle is greater than the third side]
On adding equations (i) and (ii), we get
(AB + BM + AC + CM) > 2AM
⇒ (AB + BM + MC + AC) > 2AM
⇒ AB + BC + AC > 2AM ...[∵ BC = BM + MC]
∴ Perimeter of ΔABC > 2AM
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