Question

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

Explanation:

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (two legs are equal, as indicated by the tick marks, and one right angle). In a right - isosceles triangle, the legs are of length \(x\) (let's say) and the hypotenuse \(c\) is related to the legs \(a\) and \(b\) (where \(a = b=x\)) by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Here, the hypotenuse \(c = 1\), and \(a=b = x\).

Step2: Apply Pythagorean Theorem

Substitute \(a = x\), \(b = x\) and \(c = 1\) into the Pythagorean theorem:

$$x^{2}+x^{2}=1^{2}$$

Combine like terms:

$$2x^{2}=1$$

Divide both sides by 2:

$$x^{2}=\frac{1}{2}$$

Take the square root of both sides. Since \(x\) represents a length, we take the positive square root:

$$x=\sqrt{\frac{1}{2}}$$

Step3: Rationalize the Denominator

To rationalize the denominator of \(\sqrt{\frac{1}{2}}\), we multiply the numerator and denominator inside the square root by 2:

$$x=\sqrt{\frac{1\times2}{2\times2}}=\frac{\sqrt{2}}{\sqrt{4}}=\frac{\sqrt{2}}{2}$$

Answer:

\(\frac{\sqrt{2}}{2}\)