We are given a triangle $ABC$ with the following conditions:

1. $\angle BAC = \angle CDA$
2. $AC = 8 \, \text{cm}$
3. $AB = 6 \, \text{cm}$
4. $AD = 3 \, \text{cm}$

We need to find the length of $CD$.

Step-by-step Approach:

Since $\angle BAC = \angle CDA$, triangles $ABC$ and $ACD$ are similar by the Angle-Angle (AA) similarity criterion. This is because they share the common angle $\angle A$ and the given condition that the two triangles have equal angles.

Using the property of similar triangles, the sides of the triangles are proportional:

$\frac{AB}{AC} = \frac{AD}{CD}$

Substituting the known values:

$\frac{6}{8} = \frac{3}{CD}$

Now, solve for $CD$:

$\frac{6}{8} = \frac{3}{CD}$

$CD = \frac{8 \times 3}{6} = 4 \, \text{cm}$

Final Answer:

The length of $CD$ is $4 \, \text{cm}$.

Would you like a detailed explanation of triangle similarity, or do you have any other questions?

Related Questions:

1. What is the Angle-Angle similarity criterion in triangles?
2. Can triangle similarity be applied in real-world problems?
3. How do proportional sides in similar triangles relate to their areas?
4. What other methods can be used to solve for unknown sides in triangles?
5. How does the Law of Sines differ from the concept of similar triangles?

Tip: When solving geometry problems involving similar triangles, always look for corresponding angles and sides to set up proportional relationships.