Pythagorean Triples

The expressions 2xy,x^2-y^2, and x^2+y^2, with x and y integers such that 1\leq y<x, generate Pythagorean triples. Given the triple (204,253,325), find the values for x and y that generate this triple.
Source: NCTM Mathematics Teacher 2008

SOLUTION
2xy is even
2xy=204\qquad\quad\;\; (1)
x^2-y^2=253\qquad (2)
The hypotenuse has the largest value
x^2+y^2=325\qquad (3)

Method 1
Add Eq. (2) to Eq. (3)
x^2-y^2=253
x^2+y^2=325
———————
2x^2=578
x^2=289
x=\pm 17
x=-17 is not possible because x is a positive integer
x=17
Substitute the value of x into Eq. (1)
2(17)y=204
y=6
x=17,y=6

Method 2
(x+y)^2=x^2+2xy+y^2
=325+204
=529
x+y=\pm 23
x+y=-23 is not possible because both x and y are positive integers
x+y=23\qquad (4)
x^2-y^2=(x+y)(x-y)
253=23(x-y)
x-y=11\qquad (5)
Add Eq. (4) to Eq. (5)
x+y=23
x-y=11
—————–
2x=34
x=17
Substitute the value of x into Eq. (4)
17+y=23
y=6
x=17,y=6

Answer: x=17,y=6

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About mvtrinh

Retired high school math teacher.
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