We are given that triangles $\triangle ABC$ and $\triangle DEC$ are similar, meaning their corresponding sides are proportional.

From the diagram:

• $AB = 10$, $AC = 6$, and $DE = 12$.
• We need to find the length of $CD$.

Since $\triangle ABC \sim \triangle DEC$, the ratio of corresponding sides is equal. Specifically, the ratio of $AC$ to $CD$ should be equal to the ratio of $AB$ to $DE$.

Thus, we can set up the proportion: $\frac{AC}{CD} = \frac{AB}{DE}$ Substitute the known values: $\frac{6}{CD} = \frac{10}{12}$ Simplify the ratio on the right side: $\frac{6}{CD} = \frac{5}{6}$ Now, solve for $CD$ by cross-multiplying: $6 \times 6 = 5 \times CD$ $36 = 5 \times CD$ $CD = \frac{36}{5}$ $CD = 7.2$

Thus, the length of side $CD$ is 7.2 units.

Would you like more details or have any further questions?

Here are some related questions you might find interesting:

1. How do you prove that two triangles are similar?
2. What other methods can be used to find the length of sides in similar triangles?
3. How do you calculate unknown angles in similar triangles?
4. What is the importance of proportionality in geometry?
5. How can similarity in triangles help solve real-world problems?

Tip: When working with similar triangles, always compare corresponding sides carefully to set up the correct proportions!