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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.
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.
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.
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).
Click to see the approximate measure of each of the angles in this triangle (for example, the angle $∠C$ measures approximately 36˚).
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˚.
Now you can see the triangle from the last section, with a slider to rotate it around the labeled point $A$.
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 $AB$.)
Now we’ll look at reflecting the triangle to the left across the side $\cl"red"{\ov{AB}}$.
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.
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.
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).
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.
In summary: