In the given triangle ABC if angle BAC equal to angle CDA AC equal to 8 cm AB equal to 6 cm and AD equal to 3 cm then find CD

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.

In the given triangle ABC if angle BAC equals angle CDA, AC equals 8 cm, AB equals 6 cm, and AD equals 3 cm, then find CD.

Solve for CD in a triangle using the Angle-Angle similarity theorem. This geometry problem involves similar triangles and proportional sides, with given values AC = 8 cm, AB = 6 cm, and AD = 3 cm. Suitable for students in Grades 9-10.

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