Isometries provide a dynamic way of dealing with congruency. In this chapter, we study a transformation which serves the same purpose for the notion of similarity.

Without employing photography, photocopying, or computer graphics, it is not a simple matter to produce an enlarged or reduced copy of a figure. There is, however, a physical instrument called a ** pantograph** that allows us to accomplish this.

A pantograph is formed from four thin fiat rods of equal length that are joined together by four hinge pins *P, Q, A*, and *B* so that *APBQ* is a parallelogram and *OA* = *AP*. The instrument lies flat on the drawing board and is fixed to the board at the *pivot point**O*. Pencils are attached to the instrument at points *P* and *P′*.

If an enlargement is desired, the pencil at *P* is used to trace the original feature. As this is being done, the pencil at *P′* draws a copy magnified by a factor equal to *OQ*/*OA*. If a reduction is desired, the pencil at *P′* is used to trace the figure so that the pencil at *P* draws the reduced copy.

To see why this works, note that *OAP* and *OQP′* are similar triangles by **sAs**. It follows that *O, P*, and *P′* are collinear. Moreover,

and thus the figure that is traced by *P* may be considered a “contraction” towards *O* of the figure that ...

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