Let’s transform some figures on grids.
3.1: Notice and Wonder: The Isometric Grid
What do you notice? What do you wonder?
3.2: Transformation Information
Follow the directions below each statement to tell GeoGebra how you want the figure to move. It is important to notice that GeoGebra uses vectors to show translations. A vector is a quantity that has magnitude (size) and direction. It is usually represented by an arrow.
These applets are sensitive to clicks. Be sure to make one quick click, otherwise it may count a double-click.
After each example, click the reset button, and then move the slider over for the next question.
- Translate triangle \(ABC\) so that \(A\) goes to \(A’\).
- Select the Vector tool.
- Click on the original point \(A\) and then the new point \(A’\). You should see a vector.
- Select the Translate by Vector tool.
- Click on the figure to translate, and then click on the vector.
Translate triangle \(ABC\) so that \(C\) goes to \(C’\).
- Rotate triangle \(ABC\) \(90^\circ\) counterclockwise using center \(O\).
- Select the Rotate around Point tool.
- Click on the figure to rotate, and then click on the center point.
- A dialog box will open; type the angle by which to rotate and select the direction of rotation.
- Click on ok.
- Reflect triangle \(ABC\) using line \(\ell\).
- Select the Reflect about Line tool.
- Click on the figure to reflect, and then click on the line of reflection.
Rotate quadrilateral \(ABCD\) \(60^\circ\) counterclockwise using center \(B\).
Rotate quadrilateral \(ABCD\) \(60^\circ\) clockwise using center \(C\).
Reflect quadrilateral \(ABCD\) using line \(\ell\).
Translate quadrilateral \(ABCD\) so that \(A\) goes to \(C\).
Try your own translations, reflections, and rotations.
- Make your own polygon to transform, and choose a transformation.
- Predict what will happen when you transform the image. Try it - were you right?
- Challenge your partner! Right click on any vectors or lines and uncheck Show Object. Can they guess what transformation you used?
Visit ggbm.at/eFeE2Veu for the applet.
When a figure is on a grid, we can use the grid to describe a transformation. For example, here is a figure and an image of the figure after a move.
Quadrilateral \(ABCD\) is translated 4 units to the right and 3 units down to the position of quadrilateral \(A'B'C'D'\).
A second type of grid is called an isometric grid. The isometric grid is made up of equilateral triangles. The angles in the triangles all measure 60 degrees, making the isometric grid convenient for showing rotations of 60 degrees.
Here is quadrilateral \(KLMN\) and its image \(K'L'M'N'\) after a 60-degree counterclockwise rotation around a point \(P\).
Clockwise means to turn in the same direction as the hands of a clock. The top turns to the right. This diagram shows Figure A turned clockwise to make Figure B.
Counterclockwise means to turn opposite of the way the hands of a clock turn. The top turns to the left.
This diagram shows Figure A turned counterclockwise to make Figure B.
An image is the result of translations, rotations, and reflections on an object. Every part of the original object moves in the same way to match up with a part of the image.
In this diagram, triangle \(ABC\) has been translated up and to the right to make triangle \(DEF\). Triangle \(DEF\) is the image of the original triangle \(ABC\).
A reflection across a line moves every point on a figure to a point directly on the opposite side of the line. The new point is the same distance from the line as it was in the original figure.
This diagram shows a reflection of A over line \(\ell\) that makes the mirror image B.
A rotation moves every point on a figure around a center by a given angle in a specific direction.
This diagram shows Triangle A rotated around center \(O\) by 55 degrees clockwise to get Triangle B.
A translation moves every point in a figure a given distance in a given direction.
This diagram shows a translation of Figure A to Figure B using the direction and distance given by the arrow.
A vertex is a point where two or more edges meet. When we have more than one vertex, we call them vertices.
The vertices in this polygon are labeled \(A\), \(B\), \(C\), \(D\), and \(E\).