Short Answer 
The length of segment DC in right triangle ABC, with AC as the hypotenuse and altitude BD drawn, is found to be 3 units by using the properties of similar triangles and the relationship DC/BC = BC/AC. By substituting the known lengths, we calculate DC as 36/12.
Step 1: Identify the Triangle Configuration
In right triangle ABC, we have an altitude BD drawn to the hypotenuse AC. The sides of the triangle are given as AC = 12 units and BC = 6 units. Recognizing this configuration is crucial as it will help us use properties of similar triangles to find the length of segment DC.
Step 2: Understand Similar Triangles
The triangles ABC, ADB, and BDC are all similar to each other. This similarity allows us to use the ratios of their corresponding sides. The key relationships here are:
- DC : BC corresponds to the first ratio
- BC : AC corresponds to the second ratio
By setting these ratios equal, we can express the length of segment DC in terms of the lengths of sides BC and AC.
Step 3: Solve for DC Using Ratios
Using the relationship from similar triangles, we establish the equation: DC/BC = BC/AC. Plugging in the known values gives us:
- DC = (BC²)/AC
- DC = (6 * 6)/12
- DC = 36/12
- DC = 3
Thus, the length of segment DC is determined to be 3 units.