Rigid Motions

We will now study geometry: the mathematics of shape, size, position, and measurement. We’ll start by looking at some ways to move shapes around, and how those motions affect properties of the shapes.

Translation

To the left is a simple drawing of a sailboat. It can be shifted, or translated in a flat plane, by sliding the slider below it. The red sailboat is affected by the translation, while the gray sailboat shows its original (untranslated) position.

What general direction does this translation go in: up and left, straight up, up and right, or straight right?
As the sailboat is translated, does its overall shape change or stay the same?
When the sailboat is translated, are there any points on it which stay in the same place (don’t move at all)?

If we want to be able to move the sailboat anywhere, we need to be able to translate in two different directions. Now the sliders below the boat can be used to translate it either right and left, or up and down.

How many units (small squares) do you need to translate the red boat in each direction before it completely covers the black boat?
To the right: units
Up: units

Now there is a red triangle to the left. Each side of this triangle is labeled with its approximate length (for example, the side $\ov{BC}$ is approximately 6.1 units long).

Approximately how long is the side $\ov{AC}$? units
As you slide the sliders to translate the red triangle, do the lengths of its sides change or stay the same?

Click to see the approximate measure of each of the angles in this triangle (for example, the angle $∠C$ measures approximately 36˚).

What is the approximate measure of the angle $∠B$? ˚
As you slide the slider to translate the red triangle, do its angle measurements change or stay the same?

Rotation

The sailboat is drawn to the left again. Now the slider below it rotates the sailboat around the labeled point $A$. We call counterclockwise rotations positive and clockwise rotations negative: for example, rotating by 45˚ means rotating counterclockwise by 45˚, and rotating by $-120˚$ means rotating clockwise by 120˚.

As you rotate the sailboat, does the point $A$ move or stay in the same place?
Are there any other points that always stay in the same place?
As the sailboat is rotated, does its overall shape change or stay the same?

Now you can see the triangle from the last section, with a slider to rotate it around the labeled point $A$.

Of the three labeled points $A$, $B$, $C$, which two points move as the triangle rotates? and
Which of the labeled points stays in the same place?
As you slide the slider to rotate the red triangle, do its side lengths and angle measurements change or stay the same?

Reflection

The sailboat is drawn to the left one more time. Now there is a checkbox below it to reflect it across the line $AB$ (the light blue line). (Reflecting across a line segment like $\cl"red"\ov{AB}$ means the same thing as reflecting across the entire line $AB$.)

When you reflect the sailboat, do the labeled points $\cl"red"A$ and $\cl"red"B$ move or stay in the same place?
Are there any points other than $\cl"red"A$ and $\cl"red"B$ (with or without dots) that stay in the same place?
When the sailboat is reflected, does its overall shape change or stay the same?

Now we’ll look at reflecting the triangle to the left across the side $\cl"red"{\ov{AB}}$.

Of the three labeled points $\cl"red"A$, $\cl"red"B$, $\cl"red"C$, which point moves when you reflect the triangle?
Which two labeled points stay in the same place? and
When you check the box to reflect the red triangle, do its side lengths and angle measurements change or stay the same?

Combinations of basic rigid motions

The sailboat is drawn again to the left. Now there are controls below it for all of the basic motions that we’ve looked at: translation, rotation, and reflection.

Experiment with sliding the sliders and checking or unchecking the checkbox. Is there any combination of these you can find which changes the size or shape of the sailboat?

Any combination of translations, rotations, and reflections doesn’t change the sizes and shapes of objects, so it is called a rigid motion.

Using the controls beneath the sailboat, move the red sailboat until it completely covers the black sailboat.

What rigid motion do you need to make this happen?
Click . Now what rigid motion do you need so the red sailboat completely covers the black sailboat?

To the left there are two lines, $\cl"red"{AB}$ and $\cl"red"{CD}$. You can apply a rigid motion to these lines by using the controls below them. These lines are parallel because they never cross (not even off-screen).

Experiment with sliding the sliders and checking or unchecking the checkbox. Do the red lines remain (straight) lines?
As you alter the controls, do the two lines always stay parallel, after any rigid motion?

The figure to the left shows two parallel red lines (one of which contains the points $\cl"red"A$ and $\cl"red"B$), a blue line, and two parallel green lines.

The blue line doesn't look like it crosses the green lines on the screen, but since it’s not going in the same direction as them it must cross them eventually. Alter the controls until you can see the points where the blue line crosses the green lines. What are names of those points? and
Do the two red lines stay parallel after any rigid motion?
Do the two green lines stay parallel after any rigid motion?
Is there any rigid motion which causes two differently-colored lines to become parallel?

In summary:

Any combination of translations, rotations, and reflections is called a rigid motion. Applying a rigid motion to a geometric figure has the following effects:
  • It doesn’t change the lengths of any line segments or sides, and doesn’t change the measure of any angles.
  • If the figure contains lines, they remain lines after applying the rigid motion.
  • If the figure contains parallel lines, those lines remain parallel; if it contains non-parallel lines, those lines remain non-parallel.