Question

find the length of side x in simplest radical form with a rational denominator. answer attempt 1 out of 2 x = submit answer

Explanation:

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (two legs are marked as equal, right - angled). In a right - isosceles triangle, the hypotenuse \(c\) and the legs \(a = b\) satisfy the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\). Here, \(c = 4\) and \(a=b = x\).

Step2: Apply Pythagorean Theorem

Substitute into the formula: \(4^{2}=x^{2}+x^{2}\). Simplify the right - hand side: \(16 = 2x^{2}\).

Step3: Solve for \(x^{2}\)

Divide both sides by 2: \(x^{2}=\frac{16}{2}=8\).

Step4: Solve for \(x\)

Take the square root of both sides: \(x=\sqrt{8}\). Simplify \(\sqrt{8}\) to simplest radical form: \(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\) (since \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) for \(a = 4,b = 2\) and \(\sqrt{4}=2\)).

Answer:

\(2\sqrt{2}\)