Geometer's Sketchpad

The following exercises are designed to introduce you to the Geometer's Sketchpad program. If you are familiar with this software, you can work through this very quickly. If the software is new to you, you may need to come back to these exercises again outside of class.

1. Open The Geometer's Sketchpad following the instructions given in class. Click on the banner that is displayed and you will see a blank "sketch." Notice that, along with the standard menu items along the top of the window, there is a Tool Bar along the left hand side of the sketch. From top to bottom these are the Selection Tool, the Point Tool, the Circle Tool, the Straight Object (segment, ray, line) Tool, the Label Tool, and the Information Tool.

2. Click on the Straight Object Tool and hold the mouse button down. You will see the tools for a segment, a ray, and a line appear. Slide over and select one of these by releasing the mouse button. Then move over to the sketch and use the tool to draw the corresponding object. Each of these geometric objects is determined by two points. You click and hold to set the first point, drag the mouse around and the release point determines the second point. Practice creating a few of each type of straight object.

3. Close this sketch (you do not need to save it) and open a new one. Use the File menu for these operations. Choose the Segment Tool and use it to draw a triangle. Choose the Selection Tool and use it to drag around some of the vertices of your triangle. Make sure that, while the shape changes, it remains a triangle.

   Still using the Selection Tool, click on one of the sides of the triangle. Notice how the selection is marked by the appearance of two dots on the segment. Click somewhere else on the sketch and the segment is de-selected. Hold the Shift key down while you click on all three sides. This is how you select multiple objects. Use this technique to select one side of your triangle and the opposite vertex. Then use the Parallel Line option under the Construct menu to draw a line parallel to the selected side through the selected vertex.

   Use the same technique but the Perpendicular Line option to construct a line perpendicular to one of the sides through the opposite vertex.

4. Close this sketch (without saving) and open a new one. Draw a triangle and select all three sides. Use the Point at Midpoint option from the Construct menu to construct the midpoints of each side of the triangle. Now construct the perpendicular bisectors of each side. To do this for one side, select the side and its midpoint, then use the Perpendicular Line construction. You should notice that when you drag around the vertices to change the shape of the triangle, it doesn't alter the fact that the perpendicular bisectors still all go through one point.

Translations

1. Open a new sketch. Draw a segment. Using the Label Tool (the hand) click on the two endpoints. You should see a label appear for each of these points. They will probably be A and B. If they are not, you can edit any label by double clicking on the label itself (not the labeled object) with the Label Tool.

2. Elsewhere in the sketch construct a triangle, a quadrilateral, and a circle. Now we want to create a translation that we can use to move these objects around. Use the Selection Tool to select the points A and B, in that order. Then choose the Mark Vector option from the Transform menu. You should see your first segment briefly marked in the direction from A to B. This sets us up to translate any point through the same distance and direction that it would take to move A to B. To do this, select the object(s) that you want to translate and then use the Translate option from the Transform menu.

   Select all three sides and all three vertices of the triangle and translate them. While the new sides and vertices are still selected, use the Color option from the Display menu to change the color of the new triangle. Do the same with the quadrilateral and with the circle. Now drag some of the original vertices to change the shapes of the original figures. Note how the transformed figures change correspondingly. the circle. After each translation, while

3. Using yet another color, construct the line segments that join each vertex of the original triangle to its corresponding transformed vertex. All these segments should be parallel and have the same length. They should also match your original segment AB.

4. One of the most important properties of a transformation is its set of fixed points -- the points that don't get moved by the transformation. Sketchpad provides us with a good way to see if there are any fixed points for a transformation. You create a point, transform it, then try to get the original to match up with its transformed image.

   Use the Point Tool to create a single point on your sketch. With this point selected, Translate it by our marked vector. Now use the Selection Tool to drag the original point towards its image. As you move it around, can you make it get closer to its image? You should be able to convince yourself that there are no fixed points for a translation. Save this sketch (on a floppy or in the User Folder) with a name that will help you identify it as your translation sketch.

Rotations

1. Open a new sketch. Use the Segment Tool to draw a small angle in the upper left corner, labeling the vertices something like this:

   
   Use the Selection Tool to select the three points that determine this angle. Select then in the order: outer point, vertex, other outer point (A, then B, then C). With these three points selected choose the Angle option from the Measure menu. You should see the size of the angle displayed. With the three points still selected, choose the Mark Angle option from the Transform menu. This establishes your selected angle as a candidate for a rotation.

2. As for the translation, construct a triangle and a quadrilateral that you can transform. Create a point somewhere on your sketch and label it O. Select this point and choose Mark Center from the Transform menu. Now we have set the center and the angel and we are ready to rotate!

3. Select all the vertices and sides of your triangle. Choose Rotate from the Transform menu. You should see a dialog box indicating that you are using the marked angle. This is good, Click on OK. While your transformed triangle is selected, change its color so that you can identify which one is the original. Repeat this process with the quadrilateral.

4. Use the Selection Tool to drag around the vertices of the original figures, changing their shapes an positions. Observe what happens to the corresponding transformed figures.

5. Using yet another color, join each of the original vertices to its corresponding transformed vertex. The situation looks much different from what it did with the translation. These segments no longer have the same length or the same direction. As you move around the original vertices, what happens to the colored segments?

6. Make the sketch a little less colored by selecting sides and vertices of the quadrilateral and deleting them. You should see the images disappear as well, since they are dependent on the originals. Construct the perpendicular bisector for each of the segments you constructed in the previous step. What surprising fact do you discover. This is no accident!!

7. Use another color to draw the segments from one original triangle vertex, to the center of rotation and then from the center of rotation to the transform of this vertex. Measure the size of the angle you just created. How does it compare to the angle of rotation? Save this sketch with a name that will let you identify it as your rotation sketch

Reflections

1. Open a new sketch. Use the Line Tool to draw a line anywhere in the sketch. With this line selected, choose the Mark Mirror option from the Transform menu. You should see the selection points for the line flash. Now construct a triangle and select all its vertices and sides.

   With these objects still selected, choose the Reflect option from the Transform menu. As before, change the color of the image so that you can easily identify it.

2. Drag around the original vertices an see what happens to the image. What happens if the original triangle has vertices on both sides of the mirror?

3. Use a third color to draw the segments from the original vertices to their images. Do they all have the same length? Which ones are shorter? Is there a fixed point? Is there more than one fixed point?

4. Construct the perpendicular bisectors of each of these new segments. What surprising fact do you discover? Save this sketch using an appropriate name.

Solving Some Mysteries

1. Take a moment to collect the information you learned in the previous exercises and formulate a strategy for doing this.

2. Open the sketchpad sketch Mystery1.gsp and save it to your own floppy or the User Folder. This provides what appears to be a blank sketch, but it has a mystery transformation already defined and added to the menu. Create a point anywhere in the sketch. With this point selected, choose Mystery1 from the bottom of the Transform menu. You can repeat this for as many points or segments as you like.

3. Using any means you can devise, identify this isometry. I will tell you that it is either a translation, a rotation, or a reflection. You should be able to identify which of these it is, determine the defining properties, and check out your conclusions.

4. Similarly, identify the isometries Mystery2, Mystery3, and Mystery4 defined in the corresponding sketchpad sketches.