Pythagorean Triples

Algebra Level 2

If all the solutions to the equation a 2 + b 2 = c 2 a^2+b^2=c^2 are of the form b = s 2 − t 2 2 , c = s 2 + t 2 2 b = \frac{s^2-t^2}{2}, c = \frac{s^2+t^2}{2} , then what is a a in terms of s s and t t ?

a = t + s 2 a = \frac{t+s}{2} a = ± s t a =\pm st a = s t 2 a = \frac{st}{2} a = t + s a = t+s

A P by A P , 26, USA

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2 solutions

Ralph James
Apr 22, 2016

a 2 = c 2 − b 2 a^2 = \sqrt{c^2 - b^2}

With substitution and taking the square root of both sides:

a = ( s 2 + t 2 2 ) 2 − ( s 2 − t 2 2 ) 2 a = \sqrt{\left(\frac{s^2 + t^2}{2}\right)^2 - \left(\frac{s^2 - t^2}{2}\right)^2}

After some simplification:

⟹ ( s 2 + t 2 ) 2 − ( s 2 − t 2 ) 2 4 \implies \sqrt{\frac{\left(s^2 + t^2\right)^2 - \left(s^2 - t^2\right)^2}{4}}

⟹ ( 2 s 2 ) ( 2 t 2 ) 4 ⟹ a = s t \implies \sqrt{\frac{(2s^2)(2t^2)}{4}} \implies \boxed{a = st}

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Jon Sy
May 8, 2016

Hello I think this is good problem.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I disagree guessing it was easy money

Percy 17 hax0r - 5 years ago

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