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The image you've uploaded presents a triangle with sides labeled \( a \), \( b \), \( c \), and an interior segment labeled \( x \). You are asked to express \( x \) in terms of \( a \), \( b \), and \( c \), but the given expression \( x = \frac{ac}{b} \) has been marked as incorrect.

To determine the correct expression for \( x \), we can use geometric relationships like similar triangles.

### Step-by-step explanation:

1. **Identify similar triangles:**
   The two smaller triangles within the larger triangle (formed by the line segment of length \( x \)) appear to be similar to each other because they share the same angles.

2. **Set up the proportion:**
   Since the triangles are similar, the corresponding sides will be proportional. We can set up the proportion based on the triangle sides:
   \[
   \frac{x}{b} = \frac{b}{a}
   \]
   
3. **Solve for \( x \):**
   Now, cross-multiply to find \( x \):
   \[
   x = \frac{b^2}{a}
   \]

Thus, the correct expression for \( x \) is:
\[
x = \frac{b^2}{a}
\]

This provides the relationship between \( x \), \( a \), and \( b \).

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Would you like more details or have any questions about this? Here are some related questions to explore further:

1. How do we prove the triangles are similar?
2. What are other methods for solving for \( x \) in terms of \( a \), \( b \), and \( c \)?
3. How do the Pythagorean Theorem or Law of Cosines apply in similar triangle problems?
4. Can this method be applied to other geometric shapes?
5. What would happen if \( x \) divided the triangle into two non-similar parts?

**Tip:** When solving geometric problems, always look for similar triangles, as they often provide a straightforward way to relate the sides.

Express x in terms of a, b, and c using the given triangle diagram.

Learn how to find x in terms of a, b, and c by applying the similar triangles theorem and proportional relationships. This problem covers key geometric concepts and is suitable for students in Grades 8-10.

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