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.