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.

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.
