The problem involves a triangle $PQR$ where $S$ is the midpoint of $PR$ and $T$ is the midpoint of $QR$. We are given the following equations for line segments:

• $ST = -6x + 56$
• $PQ = 10 + 5x$

Since $S$ and $T$ are midpoints, the line $ST$ is a midsegment of the triangle $PQR$. A midsegment in a triangle is parallel to the third side (in this case, $PQ$) and half of its length.

This implies: $ST = \frac{1}{2} PQ$

Step 1: Express $ST$ in terms of $PQ$

Given that $ST = \frac{1}{2} PQ$, substitute the value of $PQ = 10 + 5x$: $ST = \frac{1}{2}(10 + 5x) = 5 + \frac{5x}{2}$

Step 2: Equating the two expressions for $ST$

Now, equate this to the given equation for $ST$: $-6x + 56 = 5 + \frac{5x}{2}$

Step 3: Solve for $x$

First, multiply both sides by 2 to eliminate the fraction: $2(-6x + 56) = 2\left(5 + \frac{5x}{2}\right)$ $-12x + 112 = 10 + 5x$

Now, move all terms involving $x$ to one side and constants to the other: $-12x - 5x = 10 - 112$ $-17x = -102$

Solve for $x$: $x = \frac{-102}{-17} = 6$

Step 4: Find the measure of $ST$

Substitute $x = 6$ back into the equation for $ST$: $ST = -6(6) + 56 = -36 + 56 = 20$

Thus, the length of $ST$ is $\boxed{20}$.

Would you like more details or have any questions? Here are some follow-up questions you can explore:

1. How is the concept of a midsegment in a triangle applied to solve problems?
2. How does changing the position of the midpoints affect the length of the midsegment?
3. What would happen to $ST$ if $PQ$ had a different expression?
4. What is the general relationship between midsegments and the sides of triangles?
5. Can this method be applied to other types of triangles, such as isosceles or equilateral?

Tip: Always double-check your algebraic manipulation when working with multiple expressions for the same geometric quantity.