To find the exact length of segment $AB$, I will analyze the geometric layout based on the image.

Observations:

1. The circle is inscribed in a square, which appears to have vertices at $(0, 0)$, $(0, 6)$, $(6, 6)$, and $(6, 0)$.
2. The center of the circle is at $(3, 3)$, and the radius is $3$, since the circle fits exactly within the square.
3. Points $A$ and $B$ are marked on the boundary of the square. Point $A$ is at $(0, 0)$, and point $B$ is at $(6, 2)$.

Finding the length of segment $AB$:

The coordinates of point $A$ are $(0, 0)$ and the coordinates of point $B$ are $(6, 2)$. To find the length of the line segment, we can use the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Substitute the coordinates of $A(0, 0)$ and $B(6, 2)$:

$d = \sqrt{(6 - 0)^2 + (2 - 0)^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40}$

$d = 2\sqrt{10}$

Thus, the exact length of segment $AB$ is $2\sqrt{10}$.

Would you like more details or have any questions?

Here are some related questions:

1. How do you find the equation of the line passing through points $A$ and $B$?
2. What is the area of the square, and how does it relate to the circle's area?
3. How can we find the equation of the inscribed circle?
4. What is the slope of the line segment $AB$?
5. What is the perimeter of the square and the circumference of the circle?

Tip: When calculating the distance between two points, always use the distance formula, which is derived from the Pythagorean theorem!