Question

the triangle below is equilateral. find the length of side x in simplest radical form with a rational denominator.

Explanation:

Step1: Use properties of equilateral triangle

In an equilateral triangle, the altitude divides the triangle into two \(30 - 60-90\) right - triangles. The side lengths of a \(30 - 60 - 90\) triangle are in the ratio \(1:\sqrt{3}:2\). Here, the side of the equilateral triangle is \(2x\) (hypotenuse of the right - triangle), and the shorter leg (half of the side of the equilateral triangle) is \(6\).

Step2: Apply the ratio of \(30 - 60 - 90\) triangle

For a \(30 - 60 - 90\) triangle, if the shorter leg \(a = 6\) (opposite the \(30^{\circ}\) angle), and the hypotenuse \(c=x\) (opposite the \(90^{\circ}\) angle). Using the ratio \(c = 2a/\sqrt{3}\) (derived from \(a:b:c=1:\sqrt{3}:2\), where \(a\) is the shorter leg, \(b\) is the longer leg, and \(c\) is the hypotenuse). Substituting \(a = 6\), we get \(x=\frac{6\times2}{\sqrt{3}}\).

Step3: Rationalize the denominator

Multiply numerator and denominator by \(\sqrt{3}\): \(x=\frac{12\sqrt{3}}{3}=4\sqrt{3}\).

Answer:

\(4\sqrt{3}\)