To solve for the length of \( DE \) in quadrilateral \( BCED \) where sides \( BD \) and \( CE \) are extended to meet at point \( A \), we can apply the **segment addition** and **proportionality** of segments created by the transversal lines.
### Step 1: Analyze the Problem
We know the following lengths:
- \( BD = 18 \)
- \( BC = 8 \)
- \( CE = 2 \)
- \( AC = 7 \)
- \( AB = 3 \)
### Step 2: Find Lengths
From the information provided, we can calculate the total lengths \( AB \) and \( AC \):
- Length \( BA = AB + BD = 3 + 18 = 21 \)
- Length \( CA = AC + CE = 7 + 2 = 9 \)
### Step 3: Apply the Segment Proportionality
Since \( BD \) and \( CE \) are extended to meet at point \( A \), we can apply the segment proportion theorem for triangles created by the intersection:
From triangles \( ABE \) and \( CDE \):
\[
\frac{AB}{AC} = \frac{DE}{BC}
\]
### Step 4: Substitute Known Values
We can substitute the known lengths into the proportion:
\[
\frac{3}{7} = \frac{DE}{8}
\]
### Step 5: Solve for \( DE \)
Cross-multiply to solve for \( DE \):
\[
3 \cdot 8 = 7 \cdot DE
\]
\[
24 = 7 \cdot DE
\]
\[
DE = \frac{24}{7}
\]
Thus, the length of \( DE \) is \( \frac{24}{7} \).
### Final Answer
So, the length of \( DE \) is:
\[
\boxed{\frac{24}{7}}
\]
