Practicing Point by Point Transformations
Let's figure out some transformations.
18.1: Notice and Wonder: Obstacles
What do you notice? What do you wonder?
18.2: Obstacle Course
For each diagram, find a sequence of translations and rotations that take the original figure to the image, so that if done physically, the figure would not touch any of the solid obstacles and would not leave the diagram. Test your sequence by drawing the image of each step.
1. Take \(ABC \) to \(DEF\).
2. Take \(GHI\) to \(JKL\).
Create your own obstacle course with an original figure, an image, and at least one obstacle. Make sure it is possible to solve. Challenge a partner to solve your obstacle course.
18.3: Point by Point
For each question, describe a sequence of translations, rotations, and reflections that will take parallelogram \(ABCD\) to parallelogram \(A'B'C'D'\).
In this unit, we have been focusing on rigid transformations in two dimensions. By thinking carefully about precise definitions, we can extend many of these ideas into three dimensions. How could you define rotations, reflections, and translations in three dimensions?
Sometimes it's not hard to figure out a transformation that takes all the points of one figure directly to all the points of its image. Here, it looks like there is a 90 degree rotation that will take figure \(ABCD\) to figure \(EFGH\). It is not obvious where the center of rotation would be, though.
Instead, we could describe the transformation in 2 steps. First, translate figure \(ABCD\) by the directed line segment \(AE\). Next, rotate the image of \(ABCD\) clockwise by angle \(B'EF\) using center \(E\). It looks like this is a 90 degree rotation, but we can be sure the rotation will work if we use the labels to define the rotation instead of an angle measure. This method of matching up 1 point at a time until the whole figure has been taken to the image will work for any transformation, including ones in which it's hard to see a single transformation from one figure to the other.
A statement that you think is true but have not yet proved.
One figure is called congruent to another figure if there is a sequence of translations, rotations, and reflections that takes the first figure onto the second.
- directed line segment
A line segment with an arrow at one end specifying a direction.
If a transformation takes \(A\) to \(A'\), then \(A\) is the original and \(A'\) is the image.
- line of symmetry
A line of symmetry for a figure is a line such that reflection across the line takes the figure onto itself.
The figure shows two lines of symmetry for a regular hexagon, and two lines of symmetry for the letter I.
A reflection is defined using a line. It takes a point to another point that is the same distance from the given line, is on the other side of the given line, and so that the segment from the original point to the image is perpendicular to the given line.
In the figure, \(A'\) is the image of \(A\) under the reflection across the line \(m\).
- reflection symmetry
A figure has reflection symmetry if there is a reflection that takes the figure to itself.
- rigid transformation
A rigid transformation is a translation, rotation, or reflection. We sometimes also use the term to refer to a sequence of these.
A rotation has a center and a directed angle. It takes a point to another point on the circle through the original point with the given center. The 2 radii to the original point and the image make the given angle.
\(P'\) is the image of \(P\) after a counterclockwise rotation of \(t^\circ\) using the point \(O\) as the center.
- rotation symmetry
A figure has rotation symmetry if there is a rotation that takes the figure onto itself. (We don't count rotations using angles such as \(0^\circ\) and \(360^\circ\) that leave every point on the figure where it is.)
A figure has symmetry if there is a rigid transformation which takes it onto itself (not counting a transformation that leaves every point where it is).
A statement that has been proved mathematically.
A translation is defined using a directed line segment. It takes a point to another point so that the directed line segment from the original point to the image is parallel to the given line segment and has the same length and direction.
In the figure, \(A'\) is the image of \(A\) under the translation given by the directed line segment \(t\).