The following figure shows $12$ two dimensional points in Euclidean space,The following figure shows $12$ two$-$dimensional points in Euclidean space, e$.$g$.$

$(2,1)$ means the $1$ st point whose coordinates are $(2,1),8.(9,5)$ means

the $8$ th point whose coordinates are $(9,5).$ Use a basic hierarchical clustering

algorithm with Euclidean distance to cluster these twelve points.

$3.(3,9),4.(4,10)$

$(3,2)$

$(2,1)$

$(1)$ What is the centroid for a cluster that contains points $\{5,7,8,9,11\}$ and

why? What is the clustroid for a cluster that contains points $\{1,2,3,4\}$

and why?

$[8$ points$]$

$(2)$ Take the nearness of two clusters to be the largest distance between

any two points, one from each cluster. Plot a dendrogram showing the

clustering result. Note that the height of the dendrogram does not need

to be accurate.

$(3)$ Take the nearness of two clusters to be the average distance between all

pairs of points, one from each cluster. Plot a dendrogram showing the

clustering result. Note that the height of the dendrogram does not need

to be accurate.