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# 4.2 Reflections, Rotations and Translations

4.9   Definition (Reflections and Rotations.) We now define a family of functions from to . If , we define
 (4.10) (4.11)

Each of the eight functions just defined carries every box to another box with the same area. You should be able to see from the picture that

We can see this analytically as follows:

I will usually omit the analytic justification in cases like this.

Each of the eight functions described in definition 4.9 carries the square to itself.

4.12   Definition (Symmetry of the square.) The eight functions defined in equations (4.10)-(4.11) are called symmetries of the square.

4.13   Exercise. Let be the set shown in the figure. On one set of axes draw the sets and (label the four sets). On another set of axes draw and label the sets and .

4.14   Example. Let and let

From the picture it is clear that . An analytic proof of this result is as follows:
 (4.15) (4.16)

To show that , I need to show that (4.16) implies (4.15). This follows because

In exercise 2.18 you assumed that and have the same area. In general we will assume that if is a set and is a symmetry of the square, then and have the same area. (Cf. Assumption 5.11.)

4.17   Definition (Translate of a set.) Let be a set in and let . We define the set by

Sets of the form will be called translates of .

4.18   Example. The pictures below show some examples of translates. Intuitively each translate of has the same shape as and each translate of has the same area as .

4.19   Example (Translates of line segments.) Let . If , then

In particular , so any segment can be translated to a segment with as an endpoint.

4.20   Exercise. Let be real numbers with and . Show that

if the four question marks are replaced by suitable expressions. Include some explanation for your answer.

4.21   Exercise. Let be the set shown in the figure below.

a) Sketch the sets and .

b) Sketch the sets and , where is defined as in definition 4.9

Next: 4.3 The Pythagorean Theorem Up: 4. Analytic Geometry Previous: 4.1 Addition of Points   Index
Ray Mayer 2007-09-07