A \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle is a right isosceles triangle. If x is the length of each leg, we can use the Pythagorean Theorem to write: \(\displaystyle{x}^{{2}}+{x}^{{2}}={\left(\sqrt{2}\right)}^{{2}}\)

\(\displaystyle{2}{x}^{{2}}={2}\)

\(\displaystyle{x}^{{2}}={1}\)

\(x=1\)