Consider the incomplete paragraph proof. given: isosceles right triangle xyz (45°–45°–90° triangle) prove: in a 45°–45°–90° triangle, the hypotenuse is times the length of each leg. because triangle xyz is a right triangle, the side lengths must satisfy the pythagorean theorem, a2 + b2 = c2, which in this isosceles triangle becomes a2 + a2 = c2. by combining like terms, 2a2 = c2. which final step will prove that the length of the hypotenuse, c, is times the length of each leg?

The answer

In a 45°–45°–90° triangle, the hypotenuse is times the length of each leg. Because triangle XYZ is a right triangle, the side lengths must satisfy the Pythagorean theorem, a2 + b2 = c2, which in this isosceles triangle becomes a2 + a2 = c2. By combining like terms, 2a2 = c2. 

the  final step will prove that the length of the hypotenuse, c, is times the length of each leg
2a2 = c2 implies  c = √2a², and since a is positive, c = a √2, because 
√a² = a, 
and we know that  a = b, finally  c = a  x √2 = b  x √2