Question

decompose the figure into regions that are closest to each vertex. explain or show your reasoning.

Explanation:

Step1: Recall perpendicular - bisector property

The set of points equidistant from two points is the perpendicular - bisector of the line segment joining the two points.

Step2: Construct perpendicular - bisectors

Construct the perpendicular - bisectors of line segments \(AB\), \(BC\), \(CD\), \(DA\), \(AC\) and \(BD\).

Step3: Determine regions

The figure will be decomposed into regions by these perpendicular - bisectors. Each region will contain points that are closest to a particular vertex. For example, the region closest to vertex \(A\) will be the region bounded by the perpendicular - bisectors of \(AB\), \(AC\) and \(AD\) such that any point in this region is closer to \(A\) than to \(B\), \(C\) or \(D\).

Answer:

Construct the perpendicular - bisectors of all the line segments connecting the vertices (\(AB\), \(BC\), \(CD\), \(DA\), \(AC\), \(BD\)). The figure will be divided into regions by these perpendicular - bisectors, and each region will contain points closest to a specific vertex.