\( X Y Z \) is an equilateral triangle, the length of its side is \( 10 \sqrt{3} \mathrm{~cm} \), then the length of the diameter of its circumcircle is ............ cm . \( \begin{array}{llll}\text { (a) } 5 & \text { (b) } 10 & \text { (c) } 15 & \text { (d) } 20\end{array} \)

Given that \( XYZ \) is an equilateral triangle with the length of its side being \( 10\sqrt{3} \) cm, we need to find the length of the diameter of its circumcircle. In an equilateral triangle, the circumcircle is the circle that passes through all three vertices of the triangle. The center of the circumcircle is the circumcenter, which is the point where the perpendicular bisectors of the sides of the triangle intersect. For an equilateral triangle, the circumcenter is also the centroid of the triangle. The centroid divides each median into two segments, one of which is twice as long as the other. The longer segment is from the vertex to the centroid, and the shorter segment is from the centroid to the midpoint of the opposite side. The radius of the circumcircle is the distance from the circumcenter to any vertex of the triangle. Since the triangle is equilateral, the radius is also the distance from the circumcenter to the midpoint of any side. The diameter of the circumcircle is twice the radius. Therefore, the length of the diameter of the circumcircle is \( 2 \times 10\sqrt{3} \) cm. Let's calculate the length of the diameter of the circumcircle. Calculate the value by following steps: - step0: Calculate: \(2\times 10\sqrt{3}\) - step1: Multiply the terms: \(20\sqrt{3}\) The length of the diameter of the circumcircle is \( 20\sqrt{3} \) cm. Therefore, the correct answer is (d) 20 cm.