We are given the information:

• $DF = 9x - 39$
• The line segment shows that $DE = 47$ and $EF = 3x + 10$.

To find $EF$, we'll follow these steps:

1. The total length of $DF$ can be expressed as: $DF = DE + EF$ Substituting the known values: $9x - 39 = 47 + (3x + 10)$

2. Now, solve for $x$: $9x - 39 = 47 + 3x + 10$ Simplify: $9x - 39 = 57 + 3x$ Subtract $3x$ from both sides: $6x - 39 = 57$ Add 39 to both sides: $6x = 96$ Divide by 6: $x = 16$

3. Now that we have $x = 16$, substitute it into the expression for $EF$: $EF = 3x + 10 = 3(16) + 10 = 48 + 10 = 58$

Thus, the length of $EF$ is $58$.

Would you like any more details or have further questions?

Here are some related questions:

1. How can you verify the solution by checking $DF$?
2. What would happen if $DE$ were unknown instead of $EF$?
3. How does segment addition work geometrically in this problem?
4. Can this method be applied to any type of line segment with different expressions?
5. How does solving for $x$ affect finding other line segment values?

Tip: Always check the total length of the line to confirm your solution.