https://file.aiquickdraw.com/mathbot/file/1730210312217k5im7tdv.jpg

To solve for \( BD \), let’s break down the information given in the image.

1. **Congruent Triangles**: The problem states that \( \triangle ABC \) is congruent to \( \triangle DBC \).
   
2. **Side Lengths**:
   - \( AB = AC = 4 \) inches (given).
   - \( BC \) is a common side for both triangles.
   - \( AD = 5 \) inches (given as the length connecting \( A \) and \( D \)).

3. **Interpretation of BD**: Since \( \triangle ABC \) is congruent to \( \triangle DBC \), the line \( BD \) should be equal to \( CD \), splitting \( AD \) into two equal segments.

### Solution:
Since \( AD = 5 \) inches, and \( BD \) and \( CD \) split \( AD \) equally due to congruence:
\[
BD = \frac{AD}{2} = \frac{5}{2} = 2.5 \text{ inches}
\]

### Answer:
\[
BD = 2.5 \text{ inches}
\]

Would you like more details or have any questions?

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Here are some related questions for further practice:

1. If \( \triangle ABC \) and \( \triangle DBC \) are congruent, what can we say about the angles in both triangles?
2. How would you calculate the perimeter of \( \triangle ABC \)?
3. What is the area of \( \triangle ABC \) if it’s a right triangle?
4. Can you prove congruence using a specific triangle congruence theorem here?
5. How does the concept of symmetry apply in this diagram?

**Tip:** When triangles are congruent, corresponding sides and angles are equal. This can simplify problems by dividing measurements evenly.

Triangle ABC is congruent to Triangle DBC. Given AB = AC = 4 inches, AD = 5 inches. Find BD.

Solve for segment BD in a geometric problem where Triangle ABC is congruent to Triangle DBC. Given AB = AC = 4 inches and AD = 5 inches, this problem uses triangle congruence concepts to determine BD. Suitable for students in Grades 8-10.

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