Triangles Are Awesome!

Geometry Level 3

In a Triangle ABC, (a+b+c)(b+c-a) = k(b)(c), where k is an Integer, then greatest value of k is?
side BC represents 'a', side AC represents 'b', side AB represents 'c'.


The answer is 3.

Bhavesh Ahuja by Bhavesh Ahuja , 24, India

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2 solutions

Aziz Alasha
Jun 18, 2015

we have kbc = (b+c+a)(b+c-a) = b2+c2-a2+2bc hence k=( b2+c2-a2)/bc+2 = 2cosA+2 , if we maximize (2cosA+2 ) , the true answer for this problem is 4 i am sorry to say that ,the previous solution that k = 3 is wrong

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Bhavesh Ahuja
Feb 17, 2015

(b+c)²- a²=k(b)(c), (b²+c²-a²)=(k-2)(b)(c), (b²+c²-a²)/2(b)(c) = (k-2)/2, cosA= (k-2)/2 <1 as A≠0, k-2<2, k<4, k=3.

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