In this question, since $IJ \parallel FG$, triangles $HIJ$ and $HFG$ are similar by the AA (Angle-Angle) similarity criterion. This similarity allows us to set up a proportion between the corresponding sides of the triangles.

Given:

• $HG = 13.2$
• $IH = 9.6$
• $FH = 17.6$

We want to find the length of $HJ$.

Since $\triangle HIJ \sim \triangle HFG$, we can set up the following proportion:

$\frac{HJ}{HG} = \frac{IH}{FH}$

Substituting the known values:

$\frac{HJ}{13.2} = \frac{9.6}{17.6}$

Now, solve for $HJ$ by cross-multiplying:

$HJ = 13.2 \times \frac{9.6}{17.6}$

Calculating $HJ$:

1. Calculate $\frac{9.6}{17.6} \approx 0.5455$
2. Multiply by $13.2$:

$HJ \approx 13.2 \times 0.5455 = 7.2$

So, the length of $HJ$ is approximately 7.2.

Would you like more details on this solution or have any questions?

Here are some related questions:

1. How can you determine when two triangles are similar using only side lengths?
2. What is the significance of parallel lines in similar triangle problems?
3. How would the solution change if the triangles were not similar?
4. What other methods exist for calculating missing side lengths in similar triangles?
5. Can you explain why cross-multiplication is useful in solving proportions?

Tip: In geometry, always look for parallel lines and angle correspondences, as they often indicate similarity in triangles.