This problem involves two parallel segments, $FG \parallel IJ$, and you are asked to find the length of $IJ$, given the following information:

• $HF = 30$
• $HI = 9$
• $FG = 40$

Since the lines are parallel, we can use the concept of similar triangles. Triangles $\triangle FGH$ and $\triangle IJH$ are similar by the Basic Proportionality Theorem (also called Thales' Theorem) because $FG \parallel IJ$.

The sides of the triangles will be proportional, so the ratio of the corresponding sides is:

$\frac{IJ}{FG} = \frac{HI}{HF}$

We can plug in the given values:

$\frac{IJ}{40} = \frac{9}{30}$

Now, solve for $IJ$:

$IJ = 40 \times \frac{9}{30}$

$IJ = 40 \times \frac{3}{10} = 12$

Thus, the length of $IJ$ is 12 units.

Would you like more details or have any questions? Here are some related questions:

1. How are similar triangles identified in a problem?
2. What is the Basic Proportionality Theorem?
3. Can this problem be solved using a different method?
4. How can the concept of ratios help in geometry problems?
5. What are other properties of parallel lines in triangles?

Tip: Always look for parallel lines in triangles, as they often indicate similar triangles and proportionality.