These will be discussed in more detail as the section progresses. For three-dimensional objects in space there are only six possible types of rigid motion: translation, reflection, rotation, glide reflection, rotary reflection, and screw displacement. These isometries are called the basic rigid motions in space. An isometry in the plane moves each point from its starting position P to an ending position P, called the image of P.
It is possible for a point to end up where it started. In studying isometries, the only things that are important are the starting and ending positions. It doesn't matter what happens in between. Consider the following example: suppose you have a quarter sitting on your dresser.
In the morning you pick it up and put it in your pocket. You go to school, hang out at the mall, flip it to see who gets the ball first in a game of touch football, return home exhausted and put it back on your dresser. Although your quarter has had the adventure of a lifetime, the net result is not very impressive; it started its day on the dresser and ended its day on the dresser. Oh sure, it might have ended up in a different place on the dresser, and it might be heads up instead of tails up, but other than those minor differences it's not much better off than it was at the beginning of the day.
From the quarter's perspective there was an easier way to end up where it did. The same effect could have been accomplished by moving the quarter to its new position first thing in the morning. Then it could have had the whole day to sit on the dresser and contemplate life, the universe, and everything. If two isometries have the same net effect they are considered to be equivalent isometries.
With isometries, the? An isometry can't change a geometric figure too much. An isometry will not change the size or shape of a figure.
I can phrase this in more precise mathematical language. The image of an object under an isometry is a congruent object. An isometry will not affect collinearity of points, nor will it affect relative position of points. In other words, if three points are collinear before an isometry is applied, they will be collinear afterwards as well.
The same holds for between-ness. If a point is between two other points before an isometry is applied, it will remain between the two other points afterward. If a property doesn't change during a transformation, that property is said to be invariant. Collinearity and between-ness are invariant under an isometry. Angle measure is also invariant under an isometry. If you have two congruent triangles situated in the same plane, it turns out that there exists an isometry or sequence of isometries that transforms one triangle into the other.
So all congruent triangles stem from one triangle and the isometries that move it around in the plane. You might be tempted to think that in order to understand the effects of an isometry on a figure, you would need to know where every point in the figure is moved.
That would be too complicated. It turns out that you only need to know where a few points go in order to know where all of the points go. How many points is? With translations, for example, you only need to know the initial and final positions of one point. That's because where one point goes, the rest follow, so to speak. With isometries, the distance between points has to stay the same, so they are all kind of stuck together.
Because you will be focusing on the starting and ending locations of points, it is best to couch this discussion in the Cartesian Coordinate System.
That's because the Cartesian Coordinate System makes it easy to keep track of the location of points in the plane. When you translate an object in the plane, you slide it around. A translation in the plane is an isometry that moves every point in the plane a fixed distance in a fixed direction.
You don't flip it, turn it, twist it, or bop it. New York: Springer-Verlag, p. Gray, A. Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Unlimited random practice problems and answers with built-in Step-by-step solutions.
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