In quadrilateral \(BCED\), sides \(\overline{BD}\) and \(\overline{CE}\) are extended past \(B\) and \(C\), respectively, to meet at point \(A\). If \(BD = 18\), \(BC = 8\), \(CE = 2\), \(AC = 7\) and \(AB = 3\), then what is \(DE\)?

imsmarter1234 Oct 3, 2024

#1**+1 **

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}}

\]

LiIIiam0216 Oct 3, 2024

#2**0 **

I hope this is not too much, but can you solve it with similar triangles?

imsmarter1234
Oct 4, 2024