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, (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:

We start with the equation \(2a^{2}=c^{2}\) from the proof. To show that \(c = \sqrt{2}a\), we need to solve for \(c\) in terms of \(a\). The first option (substituting values) is not helpful as we need a general proof. The second option has an extra step of dividing by 2 which is not needed since we already have \(2a^{2}=c^{2}\). The fourth option only divides by 2, leaving \(a^{2}=\frac{c^{2}}{2}\), which doesn't give the relationship between \(c\) and \(a\) directly. The third option, taking the principal square root of both sides of \(2a^{2}=c^{2}\), gives \(\sqrt{2a^{2}}=\sqrt{c^{2}}\), which simplifies to \(a\sqrt{2}=c\) (since lengths are positive, we take the principal root), showing that the hypotenuse \(c\) is \(\sqrt{2}\) times the leg \(a\).

Answer:

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