Let's use the Pythagorean Theorem to solve this.
One side is 10 inches, the other side we don't know, and the hypotenuse is 19 inches. We will call the side we don't know x.
Thus, [tex] 10^2 + x^2 = 19^2[/tex]
[tex]x = \sqrt{19^2 - 10^2} = \sqrt{261} = \boxed{3\sqrt{29}}[/tex]
