We are given a right triangle $\triangle ABC$ with a smaller right triangle $\triangle ABD$ inside it, sharing the side $BD$. The length of side $AB$ is 5, and side $AC$ is 25. We are asked to find the length of side $BD$ in simplest radical form.

Step-by-Step Solution:

1. Identify the triangles:

   • $\triangle ABD$ is a right triangle with $\angle ADB = 90^\circ$, and $AB = 5$.
   • $\triangle ABC$ is also a right triangle with $\angle ABC = 90^\circ$, and $AC = 25$.

2. Use the Pythagorean Theorem on $\triangle ABC$: The Pythagorean Theorem is $AB^2 + BC^2 = AC^2$, so: $5^2 + BC^2 = 25^2$ Simplifying: $25 + BC^2 = 625$ $BC^2 = 600$ $BC = \sqrt{600} = 10\sqrt{6}$

3. Use similar triangles: Triangles $\triangle ABD$ and $\triangle ABC$ are similar because they share $\angle A$ and both are right triangles. Using the similarity ratio: $\frac{AB}{AC} = \frac{BD}{BC}$ Substituting the known values: $\frac{5}{25} = \frac{BD}{10\sqrt{6}}$ Simplifying: $\frac{1}{5} = \frac{BD}{10\sqrt{6}}$ Solving for $BD$: $BD = \frac{10\sqrt{6}}{5} = 2\sqrt{6}$

Thus, the length of $BD$ is $\boxed{2\sqrt{6}}$.

Would you like further details on any part of the solution?

Here are 5 relative questions for you:

1. How do you determine if two triangles are similar?
2. Can the Pythagorean Theorem be applied to any triangle, or only specific types?
3. Why do we use the ratio of sides to solve for unknown lengths in similar triangles?
4. How would you verify if two triangles are congruent instead of similar?
5. What are some other ways to find unknown sides in right triangles?

Tip: When simplifying radicals, always look for perfect squares under the square root to simplify your expression easily.