Question

find the length of side x in simplest radical form with a rational denominator.
answer attempt 2 out of 2
x = 2\sqrt{2}

Explanation:

Step1: Identify the triangle type

This is a 45 - 45 - 90 right - isosceles triangle (since the two legs are equal). In a 45 - 45 - 90 triangle, the ratio of the sides is \(1:1:\sqrt{2}\). Let the length of each of the equal sides be \(a\) and the hypotenuse be \(c\). The formula for the hypotenuse \(c=\sqrt{a^{2}+a^{2}}=\sqrt{2a^{2}} = a\sqrt{2}\). But here, we can also use the sine or cosine of 45 degrees. \(\sin45^{\circ}=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos45^{\circ}=\frac{\text{adjacent}}{\text{hypotenuse}}\), and \(\sin45^{\circ}=\cos45^{\circ}=\frac{\sqrt{2}}{2}\).

Step2: Apply the trigonometric ratio

Let the equal sides be of length \(x\). Using the cosine of 45 degrees: \(\cos45^{\circ}=\frac{x}{5}\). Since \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\), we have \(\frac{\sqrt{2}}{2}=\frac{x}{5}\).
Cross - multiply: \(2x = 5\sqrt{2}\), then \(x=\frac{5\sqrt{2}}{2}\).

Answer:

\(x = \frac{5\sqrt{2}}{2}\)