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Algebra

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8 Jan 2009 11:36:46 IST

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the no. of right angld triangles with integer sides and inradius 2009 is?

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the no. of right angld triangles with integer sides and inradius 2009 is?

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Deepak

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Joined: 19 Jan 2009

Posts: 304

1 May 2009 23:32:29 IST

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r = d/s take delta = d i.e area of triangle ABC with BC = a , CA = b and AB = c

since triangle is right at (say) C => d = 1/2ab

=> ab/2s = r => ab/(a + b + c) = r => ab / (a + b + sqrt.(a² + b²) ) = r

=> ab(a + b - sqrt(a² + b²) / 2ab = r => (a + b - sqrt(a²+ b²) ) = 2r

here r = 2009

=> a + b - sqrt(a²+ b²) ) = 4018 where a, b, c are all positive integers

now the question is converted to find number of integral solutions of the given equation

now i think u can try it out

both a and b can't be odd

so either both are even or one is even other is odd now think

Soumik

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Joined: 31 Jul 2008

Posts: 1266

2 May 2009 22:58:12 IST

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Use the concept of primitive pythagorian triplets.....

Let the sides be (s²-t²,2st,s²+t²).........

So we have 2s(s-t)=2009......meaning s divides 2009, meaning s belongs to this set (41,41,7,2009,1).........Frame the corresponding t's, and thus find the ans.......nudge if problem not clear!

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