Short Answer 
The solution involves understanding that in right triangle ABC, segment DC can be determined using similarity in triangles ABC, ADB, and BDC. By setting up the ratio of the segments, we find that the length of segment DC is 3 units.
Step 1: Understand the Triangle and its Components
In right triangle ABC, where AC is the hypotenuse and BD is the altitude to this hypotenuse, we have the following lengths:
- Hypotenuse AC = 12 units
- One leg BC = 6 units
These measurements are crucial for determining the length of segment DC. Both segments from point D to points C and B are important to establish the similarities in triangles formed.
Step 2: Identify Similar Triangles
The key to solving for DC lies in recognizing that triangles ABC, ADB, and BDC are similar. This similarity allows us to use corresponding sides to set ratios:
- Triangle ABC corresponds with triangles ADB and BDC
- Ratios we can use: DC : BC = BC : AC
This means that the ratio of the length of segment DC to the length of segment BC is the same as the ratio of the length of BC to AC.
Step 3: Set Up and Solve the Ratio Equation
Now, using the ratios established in Step 2, we can set up the equation:
- DC / BC = BC / AC
- Substituting, we have: DC / 6 = 6 / 12
From here, we find: DC = (BC^2) / AC = (6 * 6) / 12 = 3 units. Therefore, the length of segment DC is 3 units.