The problem states that triangles $\triangle ABC$, $\triangle ADC$, and $\triangle CDB$ are similar, and we are to calculate the length of segment $BD$, given that $AD = 4$ and $CD = 3$.

Step-by-step solution:

1. Understand the relationships:

   • Since the triangles $\triangle ABC$, $\triangle ADC$, and $\triangle CDB$ are similar, their corresponding sides are proportional.
   • Let the length of $BD$ be $x$.

2. Establish proportions: From triangle similarity, the ratios of corresponding sides are equal: $\frac{AD}{CD} = \frac{CD}{BD}$

3. Substitute known values: Substituting $AD = 4$ and $CD = 3$ into the proportion: $\frac{4}{3} = \frac{3}{x}$

4. Solve for $x$: Cross-multiply to find $x$: $4x = 9$ $x = \frac{9}{4}$ $x = 2.25$

5. Final answer: The length of $BD$ is: $BD = 2.25 \, \text{units}.$

Would you like further clarification on any step or a different approach?

Related Questions:

1. How do you verify that triangles are similar using angles?
2. Can this problem be solved using trigonometry instead of ratios?
3. What are the general properties of similar triangles?
4. How would the solution change if $AD$ and $CD$ were reversed?
5. How can the Pythagorean theorem be applied to verify segment lengths in this problem?

Tip:

Always label your triangles and proportions clearly to avoid confusion when solving problems involving similar triangles!