Transforming shapes without changing their areas

Under any of Euclidean transformations—reflection, rotation, and translation, the shape of a geometric object will not change; only the position and orientation of the object will change. That is, the object and its transformation are congruent. How about when the shape can change but the area remain unchanged? Try this problem.

 

Problem

How do you construct a shape that has the same area as a given shape?

Solution

Explore the following applet for hints.

The areas of the shapes do not change, it is preserved as long as they move parallel to the base. Can you explain why the area is preserved for each of the shapes? What is the significance of moving along the parallel lines?

This kind transforming shapes is called shearing. A shear is a transformation in which all points along a given line remain fixed while other points are shifted parallel to by a distance proportional to their perpendicular distance from the line. Shearing a plane figure does not change its area.

A shear is a type of affine transformation. An affine transformation is any transformation that preserves collinearity, that is, all points lying on a line initially, still lie on a line after transformation.

Extending the problem

Transform a quadrilateral into a triangle with the same area.