This might look like a rather technical and daunting task. It might also seem rather contrived, in that it relies on a specific, and particularly nice configuration, the 30˚60˚90˚ triangle. However, it is not vital that the task is solved in every detail - its prime purpose is to provide an opportunity to explore the properties of a shear.

The shear is a geometric transformation that will be relatively unfamiliar to many students and teachers. It is an affine transformation, as are the more familiar translation, rotation, reflection and enlargement.

Some properties of a shear are listed below; if they appear rather dry, then running the movies and exploring the JAVA worksheets that can be found on these pages should help give meaning to them:

A shear of the plane is defined by an invariant line and a shear factor;
all points in the plane move parallel to the invariant line;
the distance a point moves depends on the value of the shear factor and is directly proportional to the point's distance from the invariant line;
a shear preserves area;
straight lines remain straight;
parallel lines remain parallel.

We start by showing the effect of the shear M on the square ABCD: NEXT PAGE

a high-res pdf file of this new GEOaa-zz task