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In Algebra, a function is a relation in which each input value is mapped to exactly one output value. This notion of function can be extended to Geometry. However, the inputs and outputs are not going to be numbers but points.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Explore
Transforming Polygons
In the applet below, four transformations can be applied to different polygons. Each transformation will affect the polygons in different ways.
Compare the original polygon and the polygon obtained after applying a transformation. In a few words, describe the effect of each transformation.
Discussion
Transformations as Functions
Example
Translating a Polygon
Consider a transformation T that translates a polygon. Below, its effect on polygon P is shown.
Compare the side lengths and angle measures between the preimage and the image. Based on this observation, what can be said about T?
Answer
Both P and P′ have the same side lengths and angle measures. For this reason, it can be established that T does not affect the shape of P, which means that T preserves the side lengths and angle measures.
Hint
Use a ruler to find the side lengths of both polygons. To find the angle measures, use a protractor. Make a table comparing the dimensions of P and P′.
Solution
With the aid of a ruler, the side lengths of both polygons can be found.
Additionally, with the help of a protractor, the angle measures of both polygons can be determined.
It is beneficial to summarize the information about the side lengths and angle measures of both polygons in a table.
| Dimensions of P | Dimensions of P′ |
|---|---|
| AB=2.1 cm | A′B′=2.1 cm |
| BC=2.4 cm | B′C′=2.4 cm |
| CD=2 cm | C′D′=2 cm |
| AD=1.5 cm | A′D′=1.5 cm |
| m∠A=74∘ | m∠A′=74∘ |
| m∠B=92∘ | m∠B′=92∘ |
| m∠C=60∘ | m∠C′=60∘ |
| m∠D=134∘ | m∠D′=134∘ |
As it can be seen in the table, both polygons P and P′ have the same side lengths and angle measures. Therefore, it can be concluded that T does not affect the shape of P. In fact, T only affects the position of the polygon.
The transformation T preserves side lengths and angle measures.
It is important to note that the conclusion does not depend on the polygon but on the effect of T on the polygon. By transforming different polygons, the same conclusion can be obtained.
Discussion
Rigid Motions and Their Properties
Notice that the two transformations previously studied share the same property. The transformations neither affected the size nor the shape of the polygon. Still, they did affect the polygon's position on the plane. These types of transformations are called rigid motions.
Because rigid motions preserve distances, there are two properties that can be inferred from the definition.
Rule
Properties of Rigid Motions
A rigid motion preserves the side lengths and angle measures of a polygon. As a result, a rigid motion maintains the exact size and shape of a figure. Still, a rigid motion can affect the position and orientation of the figure.
Proof
- A rigid motion preserves the side lengths of a polygon because, by definition, the distance between the vertices do not change.
- It is accepted without a proof that rigid motions also preserve angle measures.
Example
Reflecting a Polygon
Ali and Magdalena are now working with a new transformation T. This transformation reflects a figure across a line as if the line was a mirror. In one of the attempts, they used a quadrilateral.
After seeing the result, Ali concluded that T is a rigid motion. However, Magdalena said it is not. Who is correct? Why does Magdalena think that T is not a rigid motion? 
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Answer
Ali is correct, T is a rigid motion. Magdalena could be confused because the transformation changes the orientation of the quadrilateral.
Solution
Using a ruler, the side lengths of both quadrilaterals can be found.
Next, with a protractor, the angle measures can be found.
In the table, all the dimensions found are summarized putting on the same row the dimensions of corresponding parts. For example, AB and A′B′ will be in the same row.
| Dimensions of ABCD | Dimensions of A′B′C′D′ |
|---|---|
| AB=2.8 cm | A′B′=2.8 cm |
| BC=2.2 cm | B′C′=2.2 cm |
| CD=1.8 cm | C′D′=1.8 cm |
| AD=1.9 cm | A′D′=1.9 cm |
| m∠A=72∘ | m∠A′=72∘ |
| m∠B=80∘ | m∠B′=80∘ |
| m∠C=88∘ | m∠C′=88∘ |
| m∠D=121∘ | m∠D′=121∘ |
As the table shows, both quadrilaterals ABCD and A′B′C′D′ have the same side lengths and angle measures. This means that T does not affect the shape of the polygon. Consequently, Ali is correct.
The transformation T is a rigid motion.
Although T is a rigid motion, notice that the orientation of the preimage and the image are not the same. In the preimage, the vertices from A to D are positioned counterclockwise, while in the image, they are positioned clockwise.
This fact could be what made Magdalena think that T is not a rigid motion. Notice that the transformations studied before preserve orientations.
Discussion
Combining Transformations
Consider the fact that two or more functions can be applied one after the other to an input. Similarly, two or more transformations can be applied one after the other to a preimage.
Concept
Composition of Transformations
A composition of transformations, or sequence of transformations, is a combination of two or more transformations. In a composition, the image produced by the first transformation is the preimage of the second transformation. The notation is similar to the notation used for functions in algebra.
Closure
Transformations in the Real World
Keep in mind that transformations transcend beyond paper and can be used for many purposes. In the real world, transformations occur everywhere. For example, in nature, lakes perform reflections of landscapes.
Also, different types of transformations can be seen by observing a car, either in rest or in movement.
- A car moving in a straight line represents a translation.
- The wheels of a moving car show rotations.
- Rear-view and side-view mirrors make reflections.
- Even car windows make reflections, and depending on the object's position, they can also make distortions.
