Geometry Help - 2 of 2

To determine the length of $DE$ in quadrilateral $BCED$, we can use the properties of similar triangles and the concept of similar triangles formed by intersecting lines.

 

Given:

- $BD = 18$

- $BC = 8$

- $CE = 2$

- $AC = 7$

- $AB = 3$

 

We need to find $DE$.

 

### Step-by-Step Solution:

 

1. **Identify triangles**:

   - Triangles $ABD$ and $ACE$ share a common vertex $A$ and extend through points $D$ and $E$ on $BD$ and $CE$, respectively.

 

2. **Use the intercept theorem (also known as Thales' theorem)**:

   According to the intercept theorem, if two lines are intersected by a pair of parallel lines, the segments intercepted on one line are proportional to the corresponding segments intercepted on the other line.

 

3. **Similar triangles**:

   Triangles $ABD$ and $ACE$ are similar by the AA criterion (angle-angle similarity) because they share angle $A$ and both have right angles.

 

4. **Set up the proportionality**:

    $$\frac{BD}{DE} = \frac{AB}{AC}$$

 

5. **Plug in the known values**:

    $$\frac{18}{DE} = \frac{3}{7}$$

 

6. **Solve for $DE$**:

    $$DE = 18 \cdot \frac{7}{3} = 18 \cdot \frac{7}{3} = 6 \cdot 7 = 42$$

 

Thus, the length of $DE$ is $\boxed{42}$.