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

Explanation:

Step1: Recall properties of equilateral triangle

In an equilateral triangle, if the side - length is (a), and we consider the right - triangle formed by the altitude. The altitude of an equilateral triangle with side - length (a) divides the triangle into two right - triangles. The hypotenuse of the right - triangle is the side of the equilateral triangle, and the base of the right - triangle is (\frac{a}{2}). Here, assume the side - length of the equilateral triangle is (a), and the given side of the right - triangle is (8) (half of the side of the equilateral triangle), and we want to find the side (x) (the altitude).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right - triangle with sides (a), (b), and hypotenuse (c), (a^{2}+b^{2}=c^{2}). Let the side of the equilateral triangle be (s), then in the right - triangle, (c = s), (a=\frac{s}{2}), and (b = x). We know that if we assume the side of the equilateral triangle is (s), then from the right - triangle formed by the altitude, (x=\sqrt{s^{2}-\left(\frac{s}{2}\right)^{2}}). Since the side of the equilateral triangle related to the right - triangle: if we consider the right - triangle with hypotenuse (s) and one side (\frac{s}{2}), and we know that the given non - hypotenuse side of the right - triangle is (8) (half of the side of the equilateral triangle), so the side of the equilateral triangle (s = 16). Then, using the Pythagorean theorem (x=\sqrt{16^{2}-8^{2}}=\sqrt{(16 + 8)(16 - 8)}=\sqrt{24\times8}=\sqrt{192}=8\sqrt{3}).

Answer:

(8\sqrt{3})