In this problem, we are given a right triangle $ABC$ with altitude $BD$ drawn to the hypotenuse $AC$, and we are asked to find the length of $BD$. It is also given that $AD = 3$ and $DC = 3$.

This problem can be solved using the geometric mean theorem (or the altitude-on-hypotenuse theorem), which states that the length of the altitude $BD$ is the geometric mean of the lengths of the two segments into which the hypotenuse is divided. Mathematically, this can be written as:

$BD^2 = AD \times DC$

Substitute the given values:

$BD^2 = 3 \times 3 = 9$

Now, solve for $BD$:

$BD = \sqrt{9} = 3$

Therefore, the length of $BD$ is 3 units.

Let me know if you need more details or have any questions! Here are 5 related questions to help you explore this topic further:

1. How can the geometric mean theorem be applied to other types of triangles?
2. What is the relationship between the sides of a right triangle when the altitude is drawn to the hypotenuse?
3. How can you use the Pythagorean theorem to find other missing lengths in right triangles?
4. What is the significance of the altitude in a right triangle in terms of triangle similarity?
5. How do you solve for the length of the hypotenuse if only the legs and altitude are known?

Tip: When working with right triangles and altitudes, always check for similar triangles to help you relate the various lengths!