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Pythagorean Triples

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Pythagorean Triples are a set of three integers that consist of a, b, c which form the sides of a right angled triangle. The smallest Pythagorean triple is the set of numbers 3, 4, and 5. Pythagorean Theorem relationship of the sides of a right triangle is attributed to the Greek mathematician and philosopher Pythagoras (ca.585-ca. 500 B.C.E.), and it bears his name (Allen G. Bulman (2005) p.471). He felt that everything could be explained by numbers (Michael Petrov 2011). Pythagorean triple is a set of three positive whole numbers a, b, and c that are the lengths of sides of a right triangle.

These formulas represent every Pythagorean triple. Given a Pythagorean Triple, we can recover d, r, and s using d = GCD (a, b, c), r = sqrt ([c + a]/ [2d]), s = sqrt ([c - a]/ [2d]). The radius is always a whole number, and is given by the formula s(r-s) d. The other formula is a+ b=c. The five Pythagorean Triples: (6,8,10) (9,12,15) (12,16,20) (15,20.25) (18,24,30). That were generated will be verified by the Pythagorean Theorem (Allen G. Bluman 2005 p.471).

(6,8,10) c=a^2+b^2

=6^2+ 8^2

=36+64

= 100

c=√100

= 10

(9,12,15 c=a^2+b^2

=9^2+ 〖12〗^2

=81+144

=225

c=√225 = 15

(12,16,20)〖 12〗^2+ 〖16〗^2=〖20〗^2

144+256

400=400

√400=40

((15,20,25) a^2+b^2=c^2

〖15〗^2+ 〖20〗^2

=225+400

=625

c =√625

=25

(18,24,30)

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