If the angles of a triangle are [tex][tex][tex]$45^{\circ}, 45^{\circ}$[/tex][/tex][/tex], and [tex][tex][tex]$90^{\circ}$[/tex][/tex][/tex], show that the length of the hypotenuse is [tex][tex][tex]$\sqrt{2}$[/tex][/tex][/tex] times as long as each leg.

1. Substitute the side lengths of the triangle into the Pythagorean theorem:
[tex]a^2 + a^2 = c^2[/tex]

2. Combine like terms:
[tex]2a^2 = c^2[/tex]