Question

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

Explanation:

Step1: Apply Pythagorean theorem

For a right - angled triangle with legs of length \(x\) and \(x\) (since the two non - right angles are \(45^{\circ}\), it's an isosceles right - triangle), and hypotenuse \(z\). By Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), here \(a = x\), \(b = x\), \(c = z\). So \(x^{2}+x^{2}=z^{2}\), which simplifies to \(2x^{2}=z^{2}\). But we can also use the property of a \(45 - 45-90\) triangle: the ratio of sides is \(1:1:\sqrt{2}\). If the hypotenuse \(z = 1\), then \(x=\frac{1}{\sqrt{2}}\).

Step2: Rationalize the denominator

Multiply numerator and denominator by \(\sqrt{2}\). \(\frac{1\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{2}}{2}\).

Answer:

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