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

Explanation:

Step1: Recall equilateral - triangle properties

In an equilateral triangle, if we draw an altitude, it bisects the base. Let the side - length of the equilateral triangle be (x). The altitude divides the equilateral triangle into two right - triangles. The base of each right - triangle is (\frac{x}{2}), the height is (\sqrt{10}), and the hypotenuse is (x).

Step2: Apply the Pythagorean theorem

In a right - triangle, (a^{2}+b^{2}=c^{2}), where (a = \frac{x}{2}), (b=\sqrt{10}), and (c = x). So ((\frac{x}{2})^{2}+(\sqrt{10})^{2}=x^{2}). Expanding, we get (\frac{x^{2}}{4}+10=x^{2}).

Step3: Solve the equation for (x)

Subtract (\frac{x^{2}}{4}) from both sides: (10=x^{2}-\frac{x^{2}}{4}=\frac{4x^{2}-x^{2}}{4}=\frac{3x^{2}}{4}). Then, cross - multiply to get (40 = 3x^{2}). So, (x^{2}=\frac{40}{3}). Taking the square root of both sides, (x=\sqrt{\frac{40}{3}}=\frac{\sqrt{40}}{\sqrt{3}}=\frac{\sqrt{4\times10}}{\sqrt{3}}=\frac{2\sqrt{10}}{\sqrt{3}}).

Step4: Rationalize the denominator

Multiply the numerator and denominator by (\sqrt{3}): (x=\frac{2\sqrt{10}\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{2\sqrt{30}}{3}).

Answer:

(\frac{2\sqrt{30}}{3})