The problem shows two similar triangles, $\triangle ABC$ and $\triangle DEF$. Since the triangles are similar, the corresponding sides of the triangles are proportional.

• For $\triangle ABC$, the sides are:

  • $AB = 6$ inches
  • $BC = 7$ inches
  • $AC$ is not given

• For $\triangle DEF$, the sides are:

  • $DE = 4$ inches
  • $DF = x$ inches
  • $EF$ is not given

To find which proportion is correct, we need to set up ratios of corresponding sides:

• $AB$ corresponds to $DE$
• $BC$ corresponds to $DF$

So, the correct proportion would be:

$\frac{AB}{DE} = \frac{BC}{DF}$

Substituting the known values:

$\frac{6}{4} = \frac{7}{x}$

This matches option D:

$\frac{4}{x} = \frac{7}{6}$

The equation can be rearranged to:

$\frac{6}{4} = \frac{7}{x}$

So the correct answer is D.

Would you like further clarification or details on the steps? Here are some follow-up questions to deepen your understanding:

1. What are corresponding sides in similar triangles?
2. How can you solve proportions like $\frac{6}{4} = \frac{7}{x}$?
3. What does it mean for two triangles to be similar?
4. How would the solution change if different sides were involved?
5. How can we apply the cross-multiplication method to solve for $x$?

Tip: Always ensure that corresponding sides are matched correctly when setting up proportions for similar triangles!