How do you prove a triangle with the hypotenuse of length 5 and other sides with lengths 3 and 4 is a right triangle?

Question

I can see that it obviously will be satisfied by the pythagorean theorem, that is:

$$3^2+4^2=5^2$$

But I am sure this isn't the way to prove the statement since you are making the assumption that it is a right triangle. I know I also can't use any of the trig functions like sin, cos, tan. Then given the sides of the triangle, can I prove this to be a right triangle?

Answer 1

If a triangle has two perpendicular sides with lengths $3$ and $4$, the length of the remaining side is $5$ by the Pythagorean theorem. Assume that a non-right triangle $T$ with side lengths $3,4,5$ exists. By the $SSS$ criterion of congruence, it is possible to overlap such triangle with the previous right triangle, contradiction.

Unwrapped version: by $SSS$, there is a unique triangle with side lenghts $3,4,5$, up to isometries. Since there is a right triangle with such side lengths, every triangle with side lenghts $3,4,5$ is a right triangle.

Alternative, creative version: by Heron's formula, the area of a triangle with side lenghts $3,4,5$ is $6$. That implies the orthogonality of the sides with lenghts $3$ and $4$, since $6=\frac{3\cdot 4}{2}$.

Answer 2

You can either use the converse of the theorem of Pythagoras or, more elegant, the cosine law