Similar Figures

by

Larry Shook

An interesting and useful tool in GSP is the ability to trace along a polygonal path. It is extremely handy in creating similar figures and performing similarity transformations. Consider the following sketches.

In each of the above figures, a segment is drawn from a point in the plane to a point on the red figure. The figure that is traced is the locus of points in blue created as the point on the red figure is animated about the red figure. The locus of points in blue is the trace of the midpoint as the animation takes place. Notice we can do this for any figure as illustrated in the following sketch.

Since the midpoint was used in each of the above sketches, the image created is 1/2 the size of the original. If we use an arbitrary point on the line, the size of the figure gets proportionally larger or smaller as the point moves along the line as shown in the next sketch.

The dotted lines illustrate how the figure will change in size as the point is moved along the segment. If the line is extended behind the point in the plane, what happens to the locus of points?

The mirror image is produced as the point is moved to the other side of the point in the plane. If you were trying to teach this topic to a class, what kind of application could you use to motivate students? Students could be asked to use this concept to draw sketches or maps to a certain scale.