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: Analyze the equation \(2a^{2}=c^{2}\)

We want to solve for \(c\) in terms of \(a\).

Step2: Take the principal square root

If \(2a^{2}=c^{2}\), then \(c = \sqrt{2a^{2}}\) (since \(c>0\) as it is the length of a side of a triangle). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a = 2\), \(b=a^{2}\)), we have \(c=\sqrt{2}\cdot\sqrt{a^{2}}\). And since \(\sqrt{a^{2}}=a\) (because \(a>0\) as it is the length of a side of a triangle), \(c = \sqrt{2}a\).

Answer:

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