We are given that $VX = 34$, $UT = 17$, and $WT$ is parallel to $VU$. We want to find the length of $TX$.
Step 2: Use the Triangle Midsegment Theorem
Since $WT$ is parallel to $VU$, $WT$ is the midsegment of $\triangle VUX$. The Triangle Midsegment Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side. Therefore, $WX = \frac{1}{2} VX$.