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In this question, we are given a triangle \( \triangle ABC \) with the following information:

- \( \overline{AB} \cong \overline{BC} \), meaning \( AB \) and \( BC \) are congruent.
- \( \overrightarrow{DB} \) is perpendicular to \( \overline{AC} \), forming a right angle at \( D \).
- \( AD = 8 \) inches.
- We are asked to find the length of \( CD \).

### Step-by-Step Solution:

Given that \( AB \cong BC \), we can conclude that \( \triangle ABC \) is an **isosceles triangle**, with \( AB = BC \).

Also, since \( \overrightarrow{DB} \) is perpendicular to \( \overline{AC} \) and the triangle is isosceles, \( D \) is the midpoint of \( \overline{AC} \) and \( BD \) is the **perpendicular bisector** of \( \overline{AC} \).

Thus, using the properties of isosceles triangles, we can state:

\[
AD = DC
\]

Since \( AD = 8 \) inches, it follows that:

\[
CD = 8 \text{ inches}.
\]

### Theorem:
The theorem used here is the **Perpendicular Bisector Theorem**, which states that in an isosceles triangle, the perpendicular drawn from the vertex angle to the base bisects the base.

### Final Answer:
- The length of \( CD \) is **8 inches**.
- The correct theorem is the **Perpendicular Bisector Theorem**.

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Would you like more details on this, or do you have any questions?

Here are 5 related questions you might explore:

1. What other properties do isosceles triangles have?
2. How do you calculate the height of an isosceles triangle?
3. What is the Converse Perpendicular Bisector Theorem?
4. How do you find the area of a triangle given its side lengths?
5. How do you prove that two sides of a triangle are congruent?

**Tip:** When dealing with isosceles triangles, remember that the altitude from the vertex always bisects the base!

Given triangle ABC with AB ≅ BC, ray DB is perpendicular to AC, AD = 8 in. What is the length of CD, and by what theorem?

Learn how to solve an isosceles triangle problem where AB ≅ BC, and a perpendicular bisector creates equal segments on the base. The length of CD is 8 inches, using the Perpendicular Bisector Theorem. Suitable for students in Grades 8-10.

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