$AD$ and $BC$ are both perpendicular to $AB$, and $CD$ is perpendicular to $AC$. If $AB = 4$ and $BC = 3$, find $CD$.

[img]https://cdn.artofproblemsolving.com/attachments/1/9/209433735d993b0d414f449d314abba3d4b90a.png[/img]
1. **Identify the right triangles and apply the Pythagorean theorem:**
   - Given that $AD$ and $BC$ are both perpendicular to $AB$, and $CD$ is perpendicular to $AC$, we can identify two right triangles: $\triangle ABC$ and $\triangle DCA$.
   - In $\triangle ABC$, we have $AB = 4$ and $BC = 3$. We need to find $AC$.
   - By the Pythagorean theorem:
     \[
     AC = \sqrt{AB^2 + BC^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
     \]

2. **Establish the similarity of triangles:**
   - Note that $\triangle ABC$ and $\triangle DCA$ are similar because they both have a right angle and share angle $A$.
   - Therefore, $\triangle ABC \sim \triangle DCA$.

3. **Set up the ratio of corresponding sides:**
   - Since $\triangle ABC \sim \triangle DCA$, the ratios of corresponding sides are equal. Specifically:
     \[
     \frac{CD}{AC} = \frac{BA}{CB}
     \]
   - Substitute the known values:
     \[
     \frac{CD}{5} = \frac{4}{3}
     \]

4. **Solve for $CD$:**
   - Cross-multiply to solve for $CD$:
     \[
     CD = 5 \cdot \frac{4}{3} = \frac{20}{3}
     \]

The final answer is $\boxed{\frac{20}{3}}$