How is the hypotenuse the longest side of any right triangle? I see that the hypotenuse of a right triangle is opposite the right angle, but how is it always the longest side?  I also know that it connects to endpoints of other sides.  Please help me out with this!  I'm really wanting to know this surprising thing.  Here's an example of a right triangle:
This is an isosceles right triangle because sides a and b (the height and the base) are the same lengths with two of the angles being 45 degrees adding up to a total with the right angle of 180 degrees (all triangles have angles that add up to 180 degrees).  I just want to know from this triangle or any other right triangles why the hypotenuse is the longest side.  You'll really be helping me out.
 A: Let $a$ be the hypotenuse and $b,c$ the others sides, then by Pythagorean Theorem
$$a^2=b^2+c^2.$$
Then
$$a^2>b^2,\quad a^2>c^2.$$
Therefore
$$a>b\quad a>c.$$
A: Alternatively, you have the law of sines:  For any triangle with sides, $A,B,C$ and corresponding angles $a,b,c$ with angle $a$ opposite side $A$ et cetera, you have the following:
$$\frac{\sin a}{A} = \frac{\sin b}{B} = \frac{\sin c}{C}$$
Let $a$ be $90$ degrees, making $A$ our hypotenuse.  Since $a+b+c = 180$ and we don't want to consider negative angles or angles equal to zero in a triangle, we have that $b<90$ and $c<90$.
$\sin a = \sin (90^\circ) = 1$
$0 < \sin b < 1$ for $0^\circ<b<90^\circ$
Using these pieces of information, you have that $B\cdot\sin a = B = A\cdot \sin b < A$
Showing that $B < A$.
Similar proof shows that $C < A$.
All of this together shows that the side opposite the 90 degree angle is the longest side in the triangle.
In a similar fashion, you can show that even for triangles which are not right triangles, the side opposite the biggest angle will be the biggest side.
A: If the angles are $\angle A \le \angle B \lt \angle C$,
then the sides are $a \le b \lt c.$
On a right triangle, two angles are acute and the third is a right angle.
Hence the side opposite the right angle is the longest side.
