We are given a right triangle $\triangle ABC$ with altitude $BD$ drawn to the hypotenuse $AC$. The following lengths are given:

• $AD = 7$,
• $BD = 14$,
  and we need to find $DC$.

Steps to Solve:

1. The Altitude Rule in a Right Triangle
   The altitude $BD$ divides the hypotenuse $AC$ into two segments, $AD$ and $DC$. The geometric mean property of a right triangle states:
   $BD^2 = AD \cdot DC$ Substituting the given values: $14^2 = 7 \cdot DC$

2. Simplify the Equation
   Calculate $14^2$: $196 = 7 \cdot DC$

3. Solve for $DC$:
   Divide both sides by 7: $DC = \frac{196}{7} = 28$

Final Answer:

The length of $DC$ is: $\boxed{28}$

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Do you want a detailed geometric explanation or a diagram?
Here are 5 related questions to deepen understanding:

1. How can we verify the altitude rule using similar triangles?
2. What is the length of the hypotenuse $AC$?
3. Can you explain the geometric mean property in detail?
4. What is the area of $\triangle ABC$?
5. How does the Pythagorean theorem relate to this setup?

Tip: Always look for relationships between segments in right triangles, as similar triangles often simplify problems.