We are given a triangle $TUV$ where:

1. $W$ is the midpoint of $TV$.
2. $X$ is the midpoint of $UV$.
3. The measure of $WX$ is given by the expression $WX = -9 + 4x$.
4. The measure of $TU$ is given by the expression $TU = -10 + 6x$.

We are asked to find the measure of $WX$.

Solution Outline

Since $W$ and $X$ are midpoints, $WX$ is a segment that connects midpoints of two sides of the triangle $TUV$. By the Midline Theorem (or Midsegment Theorem), $WX$ is parallel to $TU$ and is half of its length.

Thus: $WX = \frac{1}{2} \times TU$

Step-by-Step Solution

1. Express $WX$ in terms of $TU$: $WX = \frac{1}{2} \times (-10 + 6x)$

2. Simplify this expression: $WX = -5 + 3x$

3. Set up an equation with the given $WX$: Since $WX = -9 + 4x$, we equate this to our expression for $WX$: $-9 + 4x = -5 + 3x$

4. Solve for $x$: $4x - 3x = -5 + 9$ $x = 4$

5. Substitute $x = 4$ back into $WX$: $WX = -9 + 4(4)$ $WX = -9 + 16$ $WX = 7$

Final Answer

The measure of $WX$ is $\boxed{7}$.

Would you like more details on any step or have additional questions?

Related Questions

1. How would the solution change if $WX$ and $TU$ had different expressions?
2. Can you prove the Midline Theorem used in this problem?
3. What is the significance of midpoints in geometry, particularly in triangles?
4. How do we interpret negative values in segment expressions?
5. How would you find the lengths of $TV$ and $UV$ if needed?

Tip

Always double-check if the problem involves a midline, as it often simplifies calculations using the Midline Theorem.