For trapezoid QRST, $M$ and $P$ are midpoints of the legs. Find $x$ and $P M$, if $P M=2 x, Q R=3 x$, and $T S=10$.
\[
x=\text { type your answer... }
\]
$P M=$ type your answer...

Step 1 ：The problem involves a trapezoid and the properties of midsegments in a trapezoid. In a trapezoid, the midsegment (the segment that connects the midpoints of the two non-parallel sides) is parallel to the bases and its length is the average of the lengths of the bases. In this case, the midsegment is PM and the bases are QR and TS.

Step 2 ：We are given that PM = 2x, QR = 3x, and TS = 10. We can set up an equation using the property of the midsegment and solve for x.

Step 3 ：The equation is \( \frac{3x}{2} + 5 = 2x \)

Step 4 ：Solving this equation gives us \( x = 10 \)

Step 5 ：Once we find x, we can substitute it back into the equation PM = 2x to find PM.

Step 6 ：Substituting x into the equation gives us \( PM = 20 \)

Step 7 ：Final Answer: \( x = \boxed{10} \) and \( PM = \boxed{20} \)