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

Explanation:

Step1: Use the Pythagorean theorem

In an equilateral triangle, the height divides the triangle into two right - angled triangles. Let the side of the equilateral triangle be (x). The base of each right - angled triangle is (\frac{x}{2}), the height is (\sqrt{6}), and the hypotenuse is (x). According to the Pythagorean theorem (a^{2}+b^{2}=c^{2}), where (a = \frac{x}{2}), (b=\sqrt{6}), and (c = x). So ((\frac{x}{2})^{2}+(\sqrt{6})^{2}=x^{2}).

Step2: Simplify the equation

Expand the equation: (\frac{x^{2}}{4}+6=x^{2}). Multiply through by 4 to clear the fraction: (x^{2}+24 = 4x^{2}). Rearrange the terms: (4x^{2}-x^{2}=24), which gives (3x^{2}=24). Divide both sides by 3: (x^{2}=8). Take the square root of both sides: (x=\sqrt{8}). Simplify (\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}).

Answer:

(2\sqrt{2})