In order to fill out the table, we need to apply some basic knowledge about perpendicular and diagonal lines.

First, let's define what a perpendicular line is. A perpendicular line is a line that intersects another line at a right angle. In the case of AB, we know that it is perpendicular, which means that it will form a right angle with another line.

Next, let's look at what a diagonal line is. A diagonal line is a line that connects two opposite corners of a polygon. For AC, we know that it is a diagonal line connecting two opposite corners of the rectangle ABCD.

Since BC is the projection of the diagonal line AC, we can use this information to find its length. The projection of a line is the distance from the endpoint of the line perpendicular to the other line. In this case, the projection of AC onto BC will form a right triangle. Using the Pythagorean theorem, we can find that BC = √(AC² - AB²).

Now, let's apply this knowledge to fill out the table. For the first row, we know that AB is 24 units and BC is 15 units. Using the Pythagorean theorem, we can find that 15 = √(AC² - 24²). Solving for AC, we get an answer of 27 units.

For the second row, we know that AB is still 24 units, but now BC is 12 units. Using the Pythagorean theorem again, we can find that 12 = √(AC² - 24²). Solving for AC, we get an answer of 21 units. And for the last row, we know that AB is 4a and BC is 5a, so by substituting those values into the Pythagorean theorem, we can find that 5a = √(AC² - (4a)²). Solving for AC, we get an answer of 3√5a units.

So now we have filled out all the values for AC, BC, and AB in the table.