2023 AMC 10B Problem 24

Write the point as a sum of three independent pieces:
$$(2u - 3w,\ v + 4w) = u\,(2, 0) + v\,(0, 1) + w\,(-3, 4), \qquad u, v, w \in [0, 1].$$
So the region is obtained by starting with the segment from $(0,0)$ to $(2,0)$, sweeping it along $(0,1)$ (giving a $2 \times 1$ rectangle), and then sweeping that rectangle along $(-3, 4)$.

Sweeping a convex polygon along a vector produces a convex polygon whose boundary consists of the original edges (each appearing once on the "front" and once on the "back") together with two copies of the sweep vector. Starting from the segment $(2,0)$ and sweeping twice, every one of the three generating vectors therefore appears exactly twice on the boundary. The region is a centrally symmetric hexagon with vertices
$$(0,0),\ (2,0),\ (2,1),\ (-1,5),\ (-3,5),\ (-3,4),$$
and its side vectors in order are $(2,0), (0,1), (-3,4), (-2,0), (0,-1), (3,-4)$, with lengths $2, 1, 5, 2, 1, 5$.

Perimeter: $2(2 + 1 + 5) = 16$.

The answer is $\boxed{\textbf{(E)}\ 16}$.

A parametrization that is linear in parameters restricted to $[0,1]$ describes a Minkowski sum of segments, a zonogon. Its boundary is built from the generating segments, each traversed once in each direction, so the perimeter is simply twice the sum of the generator lengths, regardless of their directions (as long as no two are parallel). Recognizing the structure avoids any need to draw carefully.

Plug in the eight corners $(u, v, w) \in \{0,1\}^3$: $(0,0), (2,0), (0,1), (2,1), (-3,4), (-1,4), (-3,5), (-1,5)$. The region is their convex hull; $(0,1)$ and $(-1,4)$ lie inside, and the remaining six form the hexagon above. Compute the six side lengths with the distance formula and add.

• Summing the three segment lengths only once (8), or treating the region as a parallelogram spanned by just two of the segments.
• Including the interior corners $(0,1)$ or $(-1,4)$ as hexagon vertices, which produces a non-convex path and a wrong perimeter.