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Let's solve the problem using the properties of similar triangles and the Pythagorean theorem.

### Problem Recap:
We have a triangle \(BCD\) with:
- \(BC = 8\) units
- \(BD = 10\) units
- \(CD = 5\) units

Point \(N\) lies on \(BD\), and the segment \(CN\) is drawn.

### Step 1: Identify the possible relationships between the triangles.
We notice that the problem might involve the right triangle properties or similar triangles. Since we have the lengths \(BC = 8\), \(BD = 10\), and \(CD = 5\), and given that \(BD\) is the hypotenuse, it suggests that \(BCD\) is a right triangle. We can confirm this by checking:

\[
BC^2 + CD^2 = BD^2
\]
\[
8^2 + 5^2 = 10^2
\]
\[
64 + 25 = 100
\]
\[
89 \neq 100
\]

This shows \(BCD\) is not a right triangle. Therefore, we should focus on similar triangles.

### Step 2: Use the Law of Cosines to find \( \cos(\angle BCD) \).

The Law of Cosines states:
\[
BD^2 = BC^2 + CD^2 - 2 \cdot BC \cdot CD \cdot \cos(\angle BCD)
\]
Plugging in the values:
\[
10^2 = 8^2 + 5^2 - 2 \cdot 8 \cdot 5 \cdot \cos(\angle BCD)
\]
\[
100 = 64 + 25 - 80 \cdot \cos(\angle BCD)
\]
\[
100 = 89 - 80 \cdot \cos(\angle BCD)
\]
\[
11 = 80 \cdot \cos(\angle BCD)
\]
\[
\cos(\angle BCD) = \frac{11}{80}
\]

### Step 3: Apply the Law of Cosines again to find \(CN\).
Let \(CN = x\).

Using \( \angle BCN = \angle BCD \) (since \(N\) lies on \(BD\)), we can write:
\[
CN^2 = BC^2 + BN^2 - 2 \cdot BC \cdot BN \cdot \cos(\angle BCD)
\]
Since \(BN = BD - ND\) and the lengths of \(BD\) and \(ND\) are not given separately, we cannot directly apply this without knowing more about the position of \(N\). However, assuming \(N\) divides \(BD\) in a specific ratio, or if we use trigonometric properties directly, the equation might be solvable with additional data or assumptions.

### Conclusion
Since the exact division of \(BD\) by \(N\) is not provided, we stop here. The critical result is the calculation of \( \cos(\angle BCD) \). From here, further assumptions or conditions would allow the complete determination of \(CN\). If \(N\) was at the midpoint, we could have calculated using specific triangle properties.

Would you like to explore more details, or provide assumptions to further solve for \(CN\)?

Learn how to solve a triangle problem involving similar triangles and the Law of Cosines. This problem explores the properties of triangles with given side lengths and the positioning of point N on side BD. Suitable for high school students.

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