Why does a pythagorean triple give you a right angled triangle?

Question

I was thinking about the fact that people used to take a 3/4/5 length of string and use it to create a right angle when doing building, etc. But it just occurred to me that I don't know why this is guaranteed to have a right angle. Feels like this should be obvious but my searches have failed me so far.

Answer 1

Suppose that $(a,b,c)$ is a Pythagorean triple, that is, $a^2+b^2=c^2$. Then $(a+b)^2=a^2+b^2+2ab=c^2+2ab>c^2$ so $a+b>c$. Therefore, there exists a triangle whose sides are $a$, $b$ and $c$.

Now we have to prove that this triangle is right. A fast method is the Law of the Cosine which states that $$c^2=a^2+b^2-2ab\cos C$$ where $C$ is the angle opposite to the side $c$. But since $c^2=a^2+b^2$ this implies that $2ab\cos C=0$. Since $a$ and $b$ are not $0$, then $\cos C$ must be $0$ so $C$ is right.

If you don't know about this Law of the Cosine, there are other longer methods.