Midpoint Mania

What happens when you connect the midpoints of a polygon to form a new polygon?

Discussion

This investigation uses Geometers sketch pad to study the relationship of a polygon and the polygon formed by connecting the midpoints of the sides of the polygon. We will look at several different polygons and explore the relationships of area and perimeter.

Triangles Quadrilaterals Pentagon Octagon

Generalization of findings

For polygons with three sides (triangles), there seems to be greatest relationship between an original polygon and the polygon formed by connecting the midpoints. There exists a relationship between the areas and perimeters of the polygons. Several triangle properties can also be identified while exploring the triangle.

For polygons with four sides, there exists a relationship between the areas of the polygons but there is no seen relationship between the perimeters of the polygons. You will also find that a special quadrilateral is formed.

For polygons with more than four sides there does not appear to be a relationship between an original polygon and the polygon formed by connecting its midpoints.

Implications for instruction

References