In this trapezoid $RTUW$, we are given that $S$ and $V$ are midpoints of the legs $RT$ and $UW$, respectively. We need to find the value of $z$ using the given conditions:

1. $TU = z$
2. $SV = 3z - 75$
3. $RW = z + 26$

The key insight here is that $SV$, the segment connecting the midpoints of the legs $RT$ and $UW$, must be the average of the two parallel sides $TU$ and $RW$. This is a property of trapezoids: the segment connecting the midpoints of the legs is parallel to the two bases and its length is the arithmetic mean of the lengths of the bases.

Using this property, we can set up the following equation for $SV$:

$SV = \frac{TU + RW}{2}$

Substituting the values:

$3z - 75 = \frac{z + (z + 26)}{2}$

Now, let's solve this equation step by step.

First, simplify the right-hand side:

$3z - 75 = \frac{2z + 26}{2}$

Simplifying further:

$3z - 75 = z + 13$

Now, isolate $z$ by subtracting $z$ from both sides:

$2z - 75 = 13$

Add 75 to both sides:

$2z = 88$

Finally, divide by 2:

$z = 44$

So, the value of $z$ is 44.

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions to explore this concept further:

1. How can you verify that $SV$ is the average of the lengths of $TU$ and $RW$?
2. What is the significance of $S$ and $V$ being midpoints in this trapezoid?
3. How does the midpoint theorem apply in this problem?
4. Can you generalize the method used here for any trapezoid?
5. What are other properties of trapezoids involving their diagonals?

Tip: Always check the midpoint formula when midpoints are involved in trapezoid or triangle problems, as it often simplifies the relationships between sides.