Question

find the length of side $x$ in simplest radical form with a rational denominator

Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - triangle (isosceles right - triangle) because the two legs are equal (marked with the same tick). In a 45 - 45 - 90 triangle, the ratio of the sides is \(1:1:\sqrt{2}\), where the hypotenuse \(c\) is related to the leg \(a\) (or \(b\)) by \(c = a\sqrt{2}\). Here, \(c = 4\) and \(a=x\).

Step2: Solve for \(x\)

From \(c=a\sqrt{2}\), we substitute \(c = 4\) and solve for \(a\) (which is \(x\)). So, \(4=x\sqrt{2}\). Then \(x=\frac{4}{\sqrt{2}}\). Rationalize the denominator: \(x=\frac{4\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\).

Answer:

\(2\sqrt{2}\)