https://file.aiquickdraw.com/mathbot/file/1730388652025uhvhyvyr.jpg

To solve for \( DE \), given that \( D \) is the midpoint of \( \overline{CE} \), we can set up the following relationships:

Since \( D \) is the midpoint of \( \overline{CE} \), the segment \( DE \) should be half the length of \( CE \). Mathematically, we can express this as:
\[
DE = \frac{1}{2} \cdot CE
\]

We are given:
- \( DE = 2x + 9 \)
- \( CE = 10x + 9 \)

Substitute \( CE = 10x + 9 \) into the equation for \( DE \) being half of \( CE \):
\[
DE = \frac{1}{2} \cdot (10x + 9)
\]

Now, set this expression equal to the given expression for \( DE \):
\[
2x + 9 = \frac{1}{2} \cdot (10x + 9)
\]

To eliminate the fraction, multiply both sides by 2:
\[
2(2x + 9) = 10x + 9
\]
\[
4x + 18 = 10x + 9
\]

Now, solve for \( x \):
1. Subtract \( 4x \) from both sides:
   \[
   18 = 6x + 9
   \]
2. Subtract 9 from both sides:
   \[
   9 = 6x
   \]
3. Divide by 6:
   \[
   x = \frac{3}{2}
   \]

With \( x = \frac{3}{2} \), substitute back to find \( DE \):
\[
DE = 2x + 9 = 2\left(\frac{3}{2}\right) + 9 = 3 + 9 = 12
\]

So, the length of \( DE \) is:
\[
\boxed{12}
\]

Would you like further explanation, or do you have any questions?

### Related Questions:
1. How would the solution change if \( D \) were not the midpoint?
2. What if \( CE \) had a different expression?
3. Can you find \( CD \) based on the information given?
4. How would we approach this if we only knew \( CD \)?
5. Could this method be used for other geometric figures?

### Tip:
Always check if simplifying the equation at the beginning helps clarify the solution process.

D is the midpoint of CE. If DE = 2x + 9 and CE = 10x + 9, what is DE?

Solve for the length of segment DE given that D is the midpoint of CE, with DE = 2x + 9 and CE = 10x + 9. This problem involves algebraic concepts, midpoint theorem, and equation solving, suitable for Grades 8-10 students.

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