Quick geometry question

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:

\[
\boxed{\frac{24}{7}}
\]