https://file.aiquickdraw.com/mathbot/file/1725929701075buq9z6ln.jpg

The problem provides the following information:

- \( DF = 9x - 39 \)
- \( DE = 47 \)
- \( EF = 3x + 10 \)

To find \( EF \), we need to use the fact that \( DF = DE + EF \), which gives us the equation:

\[
9x - 39 = 47 + (3x + 10)
\]

Now, we will solve for \( x \).

\[
9x - 39 = 47 + 3x + 10
\]
\[
9x - 39 = 57 + 3x
\]
\[
9x - 3x = 57 + 39
\]
\[
6x = 96
\]
\[
x = \frac{96}{6} = 16
\]

Now that we have \( x = 16 \), we can substitute this value back into the expression for \( EF \).

\[
EF = 3x + 10 = 3(16) + 10 = 48 + 10 = 58
\]

Thus, the length of \( EF \) is 58 units.

Would you like any more details on the solution, or do you have any questions?

### Related Questions:
1. How would the solution change if \( DF = 9x + 39 \)?
2. How do we verify the value of \( x \) by substituting it back into the equation for \( DF \)?
3. What if \( DE \) and \( EF \) were not given in terms of expressions but just values?
4. How do we solve a similar problem where all segments are expressions?
5. What is the importance of understanding segment addition in geometry?

**Tip**: Always make sure to double-check the steps when solving for unknowns in geometric problems to avoid minor errors!

If DF = 9x - 39, find EF. Given that DE = 47 and EF = 3x + 10.

Learn how to solve a geometric problem involving algebraic expressions for segment lengths. Given DF = 9x - 39, DE = 47, and EF = 3x + 10, this problem illustrates the use of the Segment Addition Theorem and algebraic manipulation to find the value of EF. Suitable for students in Grades 8-10.

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