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? Substitute values for a and c into the original Pythagorean theorem equation. Divide both sides of the equation by two, then determine the principal square root of both sides of the equation. Determine the principal square root of both sides of the equation. Divide both sides of the equation by 2.

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 that is:

[tex]a^{2}+b^{2}=c^{2}[/tex], which in this isosceles triangle becomes [tex]a^{2}+a^{2}=c^{2}[/tex] as a=b in isosceles triangle.

By combining the like terms, [tex]2a^{2}=c^{2}[/tex]

Now, we will determine the principal square root of both sides of the equation,

[tex]c=\sqrt{2}a[/tex] (since a is positive)

Divide both sides of the equation by 2, we get

[tex]\frac{c}{2}=\frac{\sqrt{2}a}{2}[/tex]

[tex]\frac{c}{2}=\frac{a}{\sqrt{2}}[/tex]

[tex]c=\frac{2a}{\sqrt{2}}[/tex]

[tex]c=\sqrt{2}a[/tex]

Now, as a=b, then [tex]c=\sqrt{2}b[/tex]