[size=150]An [b]Isometry [/b]is defined as the transformation every pair of points, P and Q, in the plane so that the distance between them is preserved.[br][br]The isometry can be defined as a [b]Rotation[/b], [b]Reflection[/b], or [b]Translation[/b].[br][br][br][/size][center]So how can we tell what type of Isometry we have?[/center]

We can't tell anything! In fact, we do not even have an isometry. Since an isometry is always defined between two points, we don't have enough points to construct one!

In many cases, we can identify an isometry just by noticing what happens to 2 points. [b]BUT[/b], there are certain cases where the transformation may be ambiguous.[br][br][br][size=150][br][center]Can you think of any example where the 2 different isometries may transform two points the same?[/center][/size][center][size=150][/size][/center]

[size=150]Here we can see that a reflection and a rotation can sometimes be ambiguous. Reflections and rotations don't always overlap, but in the case where the [u]two points are colinear with the center of rotation[/u] the images of each isometry will overlap.[/size][br]

Here we can see that the above line segment can be a translation from left to right or a reflection over the midline. When the 2 points create line segments form parallel lines, the isometry can also be ambiguous!

[size=150]So Isometries can be ambiguous given only 2 points. [br][br][center]However, does the ambiguity continue if we add a [u]third point[/u]?[/center][/size]

[size=150]As we can see, the two polygons do [i]not[/i] overlap! This lets us know immediately whether we have a rotation or a reflection.[br][br]The [color=#6aa84f][b]green[/b] [/color]polygon can't be a reflection since C'' does not share the same midline as B' and A'.[br]Similarly, the [color=#0000ff][b]blue[/b][/color] polygon can't be a rotation since the angle [math]\angle BOC[/math] is not the same as [math]\angle B'OC'[/math][br][br][center]But is this true for all collections of 3 points?[br][/center][br][br][/size][size=150]NO! If all 3 points are collinear the "polygons" [b]will[/b] overlap! Try moving the point C onto the line AB[/size]

Similarly, the two polygons don't overlap either! Now we can tell the difference between the reflection and translation.[br][br]The [color=#6aa84f][b]green[/b] [/color]polygon can't be a reflection since the two polygons have the same orientation[br]Similarly, the [color=#0000ff][b]blue[/b][/color] polygon can't be a translation since the orientation has changed![br][br]We can further visualize the reason why there can be no overlap by considering the nature of the isometry. Isometries [i]must[/i] keep the distances between every point equal. Therefore, we can think about where each point can possibly be plot.

To keep the distance from E to C constant in the image, E' must be plot somewhere on the [b][color=#ff00ff]Pink[/color] [/b]circle.[br]To keep the distance from E to D constant in the image, E' must also be plot somewhere on the [b][color=#9900ff]Purple[/color] [/b]circle.[br][br]Therefore, the point E' must be on one of the points of intersection. There are only two different points of intersection, so once we get the third point, we only have 2 possibilities of isometries. We can tell which is which by considering the orientation.