Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change?
(a) the length of the segment $MN$
(b) the perimeter of $\triangle PAB$
(c) the area of $\triangle PAB$
(d) the area of trapezoid $ABNM$

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$
(a) Triangles $ABP$ and $MNP$ are similar, and since $PM=\frac{1}{2}AP$, $MN=\frac{1}{2}AB$.
(b) We see the perimeter changes. For example, imagine if P was extremely far to the left.
(c) The area clearly doesn't change, as both the base $AB$ and its corresponding height remain the same.
(d) The bases $AB$ and $MN$ do not change, and neither does the height, so the area of the trapezoid remains the same.
Only $1$ quantity changes, so the correct answer is $\boxed{\text{B}}$.