If the angles of a triangle are 45°, 45°, and 90°,

show that the length of the hypotenuse is #&\sqrt{\sf 2}#& times as long as each leg.

✓ 1. Substitute the side lengths of the triangle into the Pythagorean theorem.
a² + a² = c²

✓ 2. Combine like terms.
1 2 3 a² = c²

✓ 3. Take the principal square root of both sides.

#&\sqrt{{\sf2}a^2} = \sqrt{c^2}#&

✓ 4. Simplify 2 a c #&\sqrt{\sf 2}#& = 2 a c