Question

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

Explanation:

Step1: Identify the triangle type

This is a right - triangle with angles \(60^{\circ}\), \(30^{\circ}\) and \(90^{\circ}\). In a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest side (let's call it \(a\)), the side opposite \(60^{\circ}\) is \(a\sqrt{3}\) and the hypotenuse is \(2a\). Here, the hypotenuse is given as \(11\). We can also use trigonometric ratios. Let's use the sine function. The sine of an angle in a right - triangle is defined as \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). For the \(60^{\circ}\) angle, the side opposite to it is \(x\) and the hypotenuse is \(11\). So, \(\sin(60^{\circ})=\frac{x}{11}\).

Step2: Recall the value of \(\sin(60^{\circ})\)

We know that \(\sin(60^{\circ})=\frac{\sqrt{3}}{2}\).

Step3: Solve for \(x\)

Substitute \(\sin(60^{\circ})=\frac{\sqrt{3}}{2}\) into the equation \(\sin(60^{\circ})=\frac{x}{11}\). We get \(\frac{\sqrt{3}}{2}=\frac{x}{11}\). Cross - multiply to solve for \(x\): \(x = 11\times\frac{\sqrt{3}}{2}=\frac{11\sqrt{3}}{2}\).

Answer:

\(\frac{11\sqrt{3}}{2}\)