Question

In a right triangle, is the side opposite the right angle the shortest side?

Answer 1

No. It is the longest side. It can be shown by using the Sines Theorem.

Explanation:

We will show that the side opposite to the right angle is the longest side in a right triangle.

According to the Sine Theorem for every triangle we have:

`a/sinA=b/sinB=c/sinC`

Let's assume, that angle `C` is right. So the last fraction is equal to `c` (the denominator becomes `1`), so we can write that:

`a/sinA=c` and `b/sinB=c`.

So we can calculate `a` and `b` in terms of side `c`

`a=c*sinA` and `b=c*sinB`

`A` and `B` are accute angles, so we can write, that `sinA in (0;1)` and `sinB in (0;1)`, so if `sinA<1` then `a=c*sinA < c`

In the same way we can show that `b=c*sinB < c`, so finally both `a` and `b` are smaller than `c` which concludes the proof.