Since a regular hexagon can be considered to be made up of 6 equilateral triangles, a line joining the farthest vertices of a hexagon can be considered to be made up using the sides of two opposite equilateral triangles forming the hexagon. Hence, its length should be twice the side of the hexagon, in this case, 4 cm.

Now, AD divided the hexagon into two symmetrical halves. Hence, AD bisects angle D, and hence, angle ADC is  6 0 ∘ .

We can find out the value of AT using cosine formula:

**Solution:**

To find the length of AT, we can use the concept of similar triangles.

**Step 1: Draw the Hexagon**

Start by drawing a regular hexagon ABCDEF with each side measuring 2 cm.

**Step 2: Locate the Midpoint**

Locate the midpoint T on the line CD.

**Step 3: Draw Lines from T to A and C**

Draw lines from T to A and C, creating triangle ATC.

**Step 4: Identify the Properties of a Regular Hexagon**

A regular hexagon has several properties that can help us solve this problem.

- All sides of a regular hexagon are equal in length.
- Opposite sides of a regular hexagon are parallel.
- The diagonals of a regular hexagon are equal in length.

**Step 5: Identify Similar Triangles**

By drawing lines from T to A and C, we have created similar triangles: triangle ATC and triangle ABD.

- Both triangles have an angle at A.
- Both triangles have an angle at T.
- Both triangles have an angle at C.
- The corresponding sides of triangle ATC and triangle ABD are proportional.

**Step 6: Use Proportions to Solve for AT**

Since triangle ATC and triangle ABD are similar, we can set up a proportion to find the length of AT.

Let x be the length of AT.

The corresponding sides of triangle ATC and triangle ABD are AT and AB, and TC and BD, respectively.

Using the proportion:

AT/AB = TC/BD

Substituting the given values:

x/2 = 1/2

Simplifying the proportion:

x = 1

Therefore, the length of AT is 1 cm.