Question

consider the incomplete paragraph proof.
given: isosceles right triangle xyz (45°–45°–90° triangle)
prove: in a 45°–45°–90° triangle, the hypotenuse is \\(\sqrt{2}\\) times the length of each leg.

because triangle xyz is a right triangle, the side lengths must satisfy the pythagorean theorem, \\(a^2 + b^2 = c^2\\), which in this isosceles triangle becomes \\(a^2 + a^2 = c^2\\). by combining like terms, \\(2a^2 = c^2\\).

which final step will prove that the length of the hypotenuse, c, is \\(\sqrt{2}\\) 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.

Explanation:

Step1: Identify target relation

We need to derive $c = a\sqrt{2}$ from $2a^2 = c^2$.

Step2: Isolate $a^2$ term

Divide both sides of $2a^2 = c^2$ by 2:
$\frac{2a^2}{2} = \frac{c^2}{2}$
Simplify to get $a^2 = \frac{c^2}{2}$

Step3: Take principal square roots

Take principal square root of both sides:
$\sqrt{a^2} = \sqrt{\frac{c^2}{2}}$
Simplify to $a = \frac{c}{\sqrt{2}}$, then rearrange to $c = a\sqrt{2}$.

Answer:

Divide both sides of the equation by two, then determine the principal square root of both sides of the equation.