This problem involves a triangle $\triangle ABC$ with a smaller triangle $\triangle EBD$ inscribed within it. The length of $BE$ is given as 28 inches. Based on the diagram, we can deduce that $AE \parallel DC$, implying that these triangles are similar by the Basic Proportionality Theorem (also known as Thales' Theorem).

Given Information:

• $BE = 28$ inches
• $AE \parallel DC$, suggesting that $\triangle ABE \sim \triangle DBC$ by AA similarity (since they have equal corresponding angles).

Step-by-Step Solution:

Since $\triangle ABE \sim \triangle DBC$, the corresponding sides are proportional: $\frac{AB}{BD} = \frac{AE}{DC} = \frac{BE}{BC}.$

Now, based on the equal marks on the sides, the sides of $AB$, $BD$, and $DC$ appear to be congruent. If these segments are congruent, then we can assume the proportionality simplifies to 1:1. Therefore, the length of $CD$ is equal to $BE$, which is 28 inches.

Thus, $CD = 28 \text{ inches}.$

Would you like further details or clarification on any part of this solution?

Here are five related questions:

1. What is the significance of triangles being similar?
2. How does the Basic Proportionality Theorem apply in this problem?
3. Why are the ratios of the corresponding sides of similar triangles equal?
4. How would the solution change if the segments were not congruent?
5. Can this theorem be applied to all types of triangles?

Tip: Always check for parallel lines and proportional sides when solving problems involving similar triangles.