To solve this problem, we can use properties of similar triangles since the lines $\overline{EF} \parallel \overline{BC}$.

Step 1: Understand the configuration

• $\overline{EF} \parallel \overline{BC}$, so triangles $\triangle DEF$ and $\triangle DBC$ are similar by the Basic Proportionality Theorem (also called Thales' theorem).
• We are given:
  • $EF = DF - 55$
  • $DC = 32$
  • $BC = 24$

Since the triangles are similar, the corresponding sides are proportional. Therefore, we can set up the following proportion:

$\frac{EF}{BC} = \frac{DF}{DC}$

Step 2: Substitute known values

From the problem, we have the proportion:

$\frac{DF - 55}{24} = \frac{DF}{32}$

Step 3: Solve for $DF$

Cross-multiply to eliminate the fractions:

$32(DF - 55) = 24DF$

Simplify:

$32DF - 1760 = 24DF$

Now, subtract $24DF$ from both sides:

$8DF = 1760$

Solve for $DF$:

$DF = \frac{1760}{8} = 220$

Thus, the length of $\overline{DF}$ is $220$.

Final Answer:

The length of $\overline{DF}$ is $220$.

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Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How does the Basic Proportionality Theorem apply to similar triangles?
2. How can you verify if two triangles are similar using parallel lines?
3. What other methods could you use to find the length of $DF$?
4. How does the length of corresponding sides in similar triangles relate to the overall scale of the triangles?
5. Could this problem be solved using trigonometry? Why or why not?

Tip: When working with proportions in geometry, always ensure that corresponding sides are matched correctly to set up the ratio!