Tour 4: A Bestiary of Quadrilaterals

The Sketchpad applet below was built using the Custom Tool square that is built in this tour.  Every line segment is part of a square that was build using verices of preexisting squares. One vertex of the great grandparent was fixed to and animated around a circle.  All the children follow the animation.  Try clicking the Animate button and then again to stop it. Find one of the central verices that will rescale the diagram when you drag it and then animate again.
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Ideas in this Tour:


Navigating a Multi-page Document

1.      Open Quadrilaterals.gsp from the Tours folder.

This document is called a multi-page document because it contains several “pages,” just as many web sites contain many related pages.  Think of it as a sketchbook for collecting related sketches.

2.      Navigate through the sketch book using the link buttons labeled “back to TOC,” etc. and the page tables at the bottom of the window.  Use the Arrow tool to push buttons.

Exploring the Constructions

Try the following:

3.      Go to the Isosceles Trapezoid page and drag point L.
What is the relationship between points L  and K?  What makes this trapezoid isosceles?

4.      On the Rectangle page, drag points E, H and G on the rectangle (one at a time.)
How does the rectangle behave when each of these points is dragged?

5.      On the parallelogram page drag segments MN and NO (one at a time), keeping an eye on the measurements underneath.
What do you notice about angles and lengths in a parallelogram?

6.      Experiment with dragging other points and segments.  If any page gets too messy, press Undo repeatedly.

So, why do the quadrilaterals in our bestiary behave the way they do?  What holds the rectangle together?  What symmetries are work behind the controlling structures of various constructions?  This Tour is designed to help you gain some understanding of the answers to these questions for more complicated sketches.


7.      From the Isosceles Trapezoid page choose Show All Hidden from the Display menu.  Midpoint X of segment IJ and perpendicular bisector m through X should now be visible.  Experiment with dragging various parts of the sketch about.  Can you predict the behavior of drag before dragging?

8.      Deselect all other objects except line m and right click on m to choose “properties”  and produce a dialog box as shown here:

The description of the object whose properties are shown is notable.  This object was constructed using the Perpendicular Line command with X and IJ selected.  You can rehide it by checking the “Hidden” box.  Also note that it is “selectable” with the Arrow tool.

9.      Move the Properties dialog box out of the way by dragging its title bar, press the Parents pop-up menu and choose items (point X and segment IJ).  Choose one of IJ’s parents and continue navigating children and parents in the “family tree” to see how things are laid out.

Adding a Page to the Document

Here we’ll add and build a new page called square, with the basic construct for a building an invariant square.

10.   Click somewhere in Quadrilaterals.gsp to make it the active document. 

11.  On the Document Options from the File menu press the Add Page button and then choose Blank Page from the popup menu.  Title the page Square and press OK

12.  Now build the square by

§         Construct a circle and then a congruent circle centered on the circumference of the first circle by dragging the second circle’s circumference to the center of the first circle.

§         Draw a line segment connecting the circles’ centers.

§         Deselect all then select the line segment and its endpoints (clicking on each in turn) and then choose Perpendiculars from the Construct menu.

§         Deselect all and then for each circle, select the circle and the perpendicular line through its center and choose Intersections from the Construct menu (or simply ctrl+I).

§         Hide everything (select and ctrl+H) except the four vertices of the square whose edges are circle radii and construct additional segments, as necessary, to complete the square.

If you need to construct more squares, this long process could become tedious.  Almost any construction like this can be made into a “custom tool,” available from the Toolbox.

13.  With the Arrow tool on the Square page choose select all from the edit page (or ctrl+A.)

14.  Choose the Create New Tool option from the Custom Tools menu in the Toolbox.  Enter the name Square and click OK.  This will add the tool named Square to the “This Document” section of the Custom Tools menu.

15.  Experiment using the tool to draw some squares using various vertices of old squares, say, to construct new ones.  If you don’t like how it turned out try ctrl+z to undo a command.  Pay attention to how the square opens out.  Try dragging one of Adam or Eve (the original parents) to see how the rest of the construction follows.  Go crazy!

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