The problem states that segment $BD$ is the bisector of triangle $ABC$. We are given the following measurements:

• $AB = 40 \, \text{cm}$,
• $AD = 30 \, \text{cm}$,
• $CD = 12 \, \text{cm}$.

We need to find the length of side $BC$.

Solution:

Using the angle bisector theorem, which states: $\frac{AB}{BC} = \frac{AD}{CD},$ we can write: $\frac{40}{BC} = \frac{30}{12}.$

1. Simplify the fraction on the right-hand side: $\frac{30}{12} = \frac{5}{2}.$

2. Set up the proportion: $\frac{40}{BC} = \frac{5}{2}.$

3. Cross-multiply to solve for $BC$: $40 \cdot 2 = 5 \cdot BC,$ $80 = 5 \cdot BC.$

4. Divide both sides by 5: $BC = \frac{80}{5} = 16 \, \text{cm}.$

Final Answer:

The length of side $BC$ is $16 \, \text{cm}$.

Would you like more explanation or have additional questions? Here are five related questions to explore:

1. What is the angle bisector theorem and its applications in geometry?
2. How can we verify the proportionality in the angle bisector theorem with different triangle types?
3. What are other properties of triangle bisectors in relation to incenter construction?
4. How does the angle bisector theorem apply in obtuse or scalene triangles?
5. Can the angle bisector theorem be used with non-integer side lengths?

Tip: Remember, the angle bisector theorem works only with angle bisectors and ensures proportionality of the divided sides.