The problem asks for a proportion that must be true, given that triangles $\triangle ABC$ and $\triangle DEF$ are similar.

For similar triangles, corresponding sides are proportional. In this case, the side lengths of $\triangle ABC$ and $\triangle DEF$ are labeled as follows:

• $AB = 6$ inches corresponds to $DE = 4$ inches.
• $AC = 7$ inches corresponds to $DF = x$ inches.
• $BC$ (unknown, but not necessary for solving the proportions) corresponds to $EF$.

Thus, the proportion for similar triangles can be set up as: $\frac{AB}{DE} = \frac{AC}{DF}$ Substituting the given values: $\frac{6}{4} = \frac{7}{x}$

Now, you can solve for $x$:

$\frac{6}{4} = \frac{7}{x}$ $6x = 28$ $x = \frac{28}{6} = 4.67 \, \text{inches} \, (\text{rounded to two decimal places})$

Therefore, $x \approx 4.67$ inches.

Would you like more details or have any questions?

Here are 5 follow-up questions:

1. How do you determine which sides correspond in similar triangles?
2. What is the significance of the ratio of corresponding sides in similar triangles?
3. Can you find the missing side $BC$ if needed? What other information would you need?
4. How can you check if two triangles are similar without knowing all side lengths?
5. How would the problem change if it asked for the area ratio instead of side lengths?

Tip: In problems involving similar triangles, remember that the ratio of any two corresponding sides will always be equal.