triangle ABC is a right angled triangle such that Ab=BC. Bisector of angle C intersects AB at D.Prove that AC+AD=BC.

Dear Student,

Let AB = AC = a and AD = b  In a right angled triangle ABC , BC2 = AB2 + AC2 BC2 = a2 + a2 BC = a2  Given AD = b, we get  DB = AB  AD or DB = a  b  We have to prove that AC + AD = BC or (a + b) = a2.  By the angle bisector theorem, we get  ADDB =ACBCb(a - b) =aa2 ba-b=12b= (a  b)2 b2=a-bb2+b=ab2+1=ab=a2+1 Rationalizing the denominator with (2-1)   b = a2+1×2-12-1  b =a2-1 2-1 b =a2-1   b = a2  a  b + a = a2  or  AD + AC = BC    [we know that AC = a, AD = b and BC = a2]  Hence it is proved.

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