Determine the side lengths of a right triangle if they are integers and the product of the legs’ lengths equals three times the perimeter.

Asked Feb 16 '23 at 15:59

Active Jul 21 '23 at 02:54

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I tried this question for 3+ hours and I get 6 and 0 every time. Stuck now and don’t know how to approach it.

I did it like this:- Let x and y be the base and height of right triangle respectively. Then $xy = 3(x + y + \sqrt {x^2 + y^2})$

$xy - 3x - 3y = 3\sqrt {x^2 + y^2}$

$xy - 3x - 3y + 9 = 3\sqrt {x^2 + y^2} + 9$

$(x-3)(y-3) = 3(\sqrt {x^2 + y^2} + 3)$

then comparing these two products with every possible combination.

This question is from Number Theory by Titu and Dorin Page 160

edited Jul 21 '23 at 02:54

user1176409

asked Feb 16 '23 at 15:59

user1150809

4

Just look at art of problem solving. Your equation gives $(x-6)(y-6)=18$. – Dietrich Burde Feb 16 '23 at 16:04
Thanks! I did it in a different way, let me add that. – user1150809 Feb 16 '23 at 17:09
As per first comment, square, cancel $xy,,$ then complete the rectangle as in the linked dupe. – Bill Dubuque Feb 16 '23 at 18:08
1

All posts on this site are questions, and all questions about diophantine equations are number theory questions – so your title is doubly redundant. – joriki Feb 16 '23 at 18:10

0 Answers0