In the diagram below, start overline, E, F, end overline 
EF
  is parallel to start overline, B, C, end overline 
BC
 . If E, FEF is 55 less than D, FDF, D, C, equals, 32DC=32, and B, C, equals, 24BC=24, find the length of start overline, D, F, end overline 
DF
 . Figures are not necessarily drawn to scale. State your answer in simplest radical form, if necessary.

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 \( 220 \).

---

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!

In the diagram below, EF is parallel to BC. If EF is 55 less than DF, DC = 32, and BC = 24, find the length of DF. Figures are not necessarily drawn to scale. State your answer in simplest radical form, if necessary.

Solve a geometry problem involving similar triangles and the Basic Proportionality Theorem. Given EF is parallel to BC, and side lengths EF, DF, and BC, learn how to find the length of DF using proportions. Suitable for students in Grades 9-11.

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