We need to calculate the measure of $DE$, where we know the following:

• $GH = -8x + 83$,
• $DE = -2x + 54$,
• $G$ and $H$ are midpoints of $\overline{DF}$ and $\overline{EF}$, respectively.

Step 1: Midsegment theorem

By the midsegment theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Therefore: $GH = \frac{1}{2} DE$

Step 2: Relate $GH$ and $DE$

Using the equation $GH = \frac{1}{2} DE$, substitute the expressions for $GH$ and $DE$: $-8x + 83 = \frac{1}{2}(-2x + 54)$

Step 3: Simplify the equation

Multiply both sides by 2 to eliminate the fraction: $2(-8x + 83) = -2x + 54$ $-16x + 166 = -2x + 54$

Step 4: Solve for $x$

Rearrange the equation to isolate $x$: $-16x + 2x = 54 - 166$ $-14x = -112$ $x = 8$

Step 5: Calculate $DE$

Now that we know $x = 8$, substitute it into the expression for $DE$: $DE = -2x + 54$ $DE = -2(8) + 54$ $DE = -16 + 54$ $DE = 38$

Final Answer:

The measure of $\overline{DE}$ is: $\boxed{38}$

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Would you like further details or have any questions? Here are some related questions for exploration:

1. What is the length of $GH$ given $x = 8$?
2. How does the midsegment theorem apply to other triangles?
3. What happens if the coordinates of $D, E,$ and $F$ are given? How would we calculate $GH$ and $DE$?
4. Can we verify the parallel nature of $GH$ and $DE$ geometrically?
5. How would the solution change if $GH = -6x + 50$ instead?

Tip: The midsegment theorem is a powerful tool in geometry—always relate midsegments to their corresponding triangle sides!