Inverting Geometric Figures

Once you can construct the inverse of a given point P with respect to gamma, you can turn your attention to inverting entire geometric figures. The Inverting Geometric Figures Menu on the previous page contains links to four files created with Geometer's Sketchpad. These files give the user the ability to animate P along the path of the figure to be inverted and traces the path of P'. By clicking on the page tabs at the bottom of each sketch, the user can explore different sets of given conditions. These sketches can be used by teachers and students as a way to make and test their conjectures. Questions and activities for students are listed below.

Circles

1. For each of the provided sketches, make conjectures about the path of P'. Test and revise your conjectures by animating P.

2. How does the position of the circle with respect to gamma affect its inverse?

3. How does the size of the circle with respect to gamma affect its inverse?

Lines

1. For each of the provided sketches, make conjectures about the path of P'. Test and revise your conjectures by animating P.

2. How does the position of the line with respect to gamma affect its inverse?

More Circles

1. For each of the provided sketches, make conjectures about the path of P'. Test and revise your conjectures by animating P.

Other Conic Sections

1. For each of the provided sketches, make conjectures about the path of P'. Test and revise your conjectures by animating P.

2. How does the position of the conic section with respect to gamma affect its inverse?

3. How does altering the shape (by changing the focal distance) of each conic section affect its inverse?

In some cases, one of these sketches could occupy your students for an entire lesson. Others are merely worth a look before moving on. The depth at which you direct your students to investigate depends upon your assessment of your students' needs, abilities, and interest. If your students have a strong background, they may wish to verify their conjectures using analytic geometry.