= Solution
{c}
With two distinct <centroids>, <K-means clustering> assigns points on opposite sides of their <nearest-centroid decision hyperplane>, which in two dimensions is a straight <line>. This constraint explains the typical outcomes in the three source plots; exact assignments depend on initialization and the realized points.
\b[Dataset A:] one <cluster in cluster analysis> contains most of the horizontal cloud and the bottom of the vertical cloud; the other contains the upper part of the vertical cloud. The two visually elongated groups are poorly matched to squared <Euclidean distance> from their <centroids>. Points at the lower end of the vertical group can be closer to the horizontal group's <centroid>, so the two original shapes need not be recovered intact.
\b[Dataset B:] a roughly straight cut divides the ring and generally the central cloud between the two <clusters in cluster analysis>. A <Voronoi cell> for either <centroid> is a <half-plane>; it cannot select a surrounding ring while excluding its center. The cut's direction can vary because the geometry is approximately rotationally symmetric.
\b[Dataset C:] the two dense lower clouds typically receive different <centroids>. The much smaller upper cloud is absorbed into one or divided between both <clusters in cluster analysis>, according to the nearly vertical separating <line>. With $K=2$, the objective usually benefits more from separating the two large clouds than from spending one <centroid> on the sparse third group. These are typical qualitative outcomes, rather than guarantees about every initialization.
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