To solve the problem of finding the length of side CD in the regular hexagon ABCDEF, we first need to understand the properties of a regular hexagon and how to represent its vertices in the coordinate plane. A regular hexagon has equal sides and equal angles, and its vertices can be represented using complex numbers or coordinates based on their positions.

Understanding the Hexagon's Structure

In a regular hexagon, each internal angle measures 120 degrees, and the sides are all of equal length. Given that side AB is represented as 4i, we can interpret this as the coordinates of point A being (0, 0) and point B being (0, 4) in the Cartesian plane. This means that side AB is vertical and has a length of 4 units.

Finding Other Vertices

To find the coordinates of the other vertices, we can use the properties of rotation in the plane. Each vertex can be found by rotating the previous vertex by 60 degrees (since a hexagon has 6 sides, and 360 degrees divided by 6 equals 60 degrees). We can calculate the positions of the other vertices as follows:

• Point A: (0, 0)
• Point B: (0, 4)
• Point C: Rotate B by 60 degrees:

Using the rotation formula, the coordinates of point C can be calculated as:

C = (x cos θ - y sin θ, x sin θ + y cos θ) where θ = 60 degrees.

Substituting the coordinates of B (0, 4):

C = (0 cos 60 - 4 sin 60, 0 sin 60 + 4 cos 60) = (0 - 4 * √3/2, 0 + 4 * 1/2) = (-2√3, 2).

Continuing to Find D and E

Next, we find point D by rotating point C by another 60 degrees:

D = (C_x cos 60 - C_y sin 60, C_x sin 60 + C_y cos 60).

Substituting C's coordinates:

D = (-2√3 cos 60 - 2 sin 60, -2√3 sin 60 + 2 cos 60) = (-2√3 * 1/2 - 2 * √3/2, -2√3 * √3/2 + 2 * 1/2) = (-√3 - √3, -3 + 1) = (-2√3, -2).

Finding CD

Now that we have the coordinates of points C and D, we can find the length of side CD. The coordinates of C are (-2√3, 2) and those of D are (-2√3, -2). The length of CD can be calculated using the distance formula:

Distance = √[(x2 - x1)² + (y2 - y1)²].

Substituting the coordinates of C and D:

Distance CD = √[(-2√3 - (-2√3))² + (-2 - 2)²] = √[0 + (-4)²] = √[16] = 4.

Final Result

Thus, the length of side CD in the regular hexagon ABCDEF is 4 units. This matches the length of all other sides, confirming the regularity of the hexagon.