Is it a triangle?

Geometry Level 3

Let x , y x, y be positive integers such that a triangle with the side lengths of 25 x 2 + 16 y 2 , 2 4 x 2 + y 2 , 81 x 2 + 4 y 2 \sqrt{25x^2+16y^2} , 2\sqrt{4x^2+y^2}, \sqrt{81x^2+4y^2} has area 39 39 .

Find the total possible numbers of ordered pairs ( x , y ) (x, y) .


The answer is 2.

Alan Yan by Alan Yan , 20, USA

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1 solution

Alan Yan
Aug 21, 2015

You can embed the triangle in this rectangle. The area of the triangle can be expressed as 36 x y − 10 x y − 4 x y − 9 x y = 13 x y = 39 ⟹ x y = 3 36xy - 10xy - 4xy -9xy = 13xy = 39 \implies xy = 3 Therefore the two solutions are (3, 1) and (1, 3).

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