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 √2 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 √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: Analyze the given equation

We have \(2a^{2}=c^{2}\). Our goal is to get \(c\) in terms of \(a\).

Step2: Take square - root of both sides

To isolate \(c\), we take the principal square root of both sides of the equation \(2a^{2}=c^{2}\). We know that \(\sqrt{c^{2}} = c\) (since \(c\) represents the length of a side and is non - negative) and \(\sqrt{2a^{2}}=\sqrt{2}\times\sqrt{a^{2}}=\sqrt{2}a\).

Answer:

C. Determine the principal square root of both sides of the equation.