12. M is a point on side BC of a AABC such that
AM is the bisector of ZBAC. Is it true to say
that perimeter of the triangle is greater than
2AM? 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 angle BAC.
In ΔABM, AB + BM > AM ........(I)
[as, sum of two sides of a triangle is greater than the third side]
In ΔACM, AC + CM > AM ..........(ii)
[as, sum of two sides of a triangle is greater than the third side]
On adding Eqs. (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 Hence proved.
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