Solution: To solve this problem, we need to use the theorem equal chords in a circle are equidistant from the center. This means that the segments AB and CD have the same distance from the center O, and since AO = OD, they must also have the same length.

Now, we can use the Pythagorean theorem to find the length of the line segment BD. Let's assume that BD = x, then we can set up the following equations:

AC² = AB² + BC²

CD² = CB² + BD²

Since AB = CD and BC = CB, we can simplify the equations to:

AC² = 2x²

CD² = 2x²

Subtracting these two equations, we get:

AC² - CD² = 0

Using the factorization formula for the difference of squares, we can rewrite this as:

(AC + CD)(AC - CD) = 0

Solving for x, we get x = AC = CD = 5.3. Therefore, the length of BD is also 5.3.