Part 1:

• What are the characteristics of an isosceles triangle? What properties of geometric figures do you know that might be of help to you to construct an isosceles triangle and guarantee that it is isosceles without using any measurement tools?  Hint: Recall the characteristics of isosceles triangles and how they might be related to properties of a circle.

• On paper, use a compass to construct a circle. Use the circle and a straightedge to construct two congruent segments with a common endpoint. Without measuring, how can you guarantee these segments are congruent? From these segments, construct an isosceles triangle.

• Now in a similar way, construct an isosceles triangle with Sketchpad.

• Verify that the triangle you constructed is isosceles by checking the lengths of the triangle’s sides.  Measure the triangle’s three interior angles, and classify your isosceles triangle.  
   
  

• Investigate the sum of the measures of the three interior angles.  Drag any vertex to manipulate your triangle. Observe what happens to the posted lengths and angle measurements.  What conclusions can you draw from your observations?  Discuss and prove a conjecture about angle measurement that describes your findings.  
  
  

• Compare and contrast the effects of dragging each of the triangle’s vertices around your sketch window. How are the ways you may manipulate your triangle related to your choice of vertex?
   

• Discuss the differences between drawing and constructing a triangle.  
  
  

• Save your sketch onto a disk as isosceles.gsp.  Keep the sketch open on your computer! (Instructor Note: A sketchpad script illustrating the construction of an isosceles triangle has been saved as isostri.gss.)

Part 2:

• What are the characteristics of an equilateral triangle?  Drag one of the vertices of your constructed isosceles triangle until the triangle appears to be equilateral.   
   

• Without using measurement tools, how can you confirm that this third side is congruent to the two radii of the circle?  
   

• Construct an equilateral triangle that is guaranteed to always be equilateral without the use of any measurement tools. 
  Hint: The Point of Intersection command under the Construct menu.  
  
  

• Click and drag one of the vertices of your triangle.  How does it affect the equilateral triangle?  
  
  

• Verify that the triangle is equilateral by measuring the length of the sides and the interior angles of the triangle. Now drag one of the vertices.  Observe what happens to the posted lengths and angle measurements. Write, discuss and prove a conjecture describing your findings.

Part 3:

Scripts are generalized recordings of sketches.  It is sometimes convenient to record the steps of particular sketches (e.g., regular polygons) that you have made for subsequent playback. For example, when a future sketch requires a constructed equilateral triangle, you can just play back your script rather than reconstructing one. Sketchpad can generate a script to record a construction while you are sketching it.  Since scripts are savable, they become a part of your basic geometric “bag of tricks”.  You can use scripts repeatedly to generate figures, or portions of figures, while sketching. Individual scripts can be used as foundations for larger scripts, leading to potentially more complex constructions.  Each script consists of a given set of geometric objects and a sequence of steps to generate a sketch.  

• Record a script of the construction of an equilateral triangle using the construction procedure you used earlier. (Instructor Note: A sketchpad script illustrating the construction of an equilateral triangle has been saved as equil.gss.)
  

• To finish your sketch, select any objects you used during the construction process that you do not want in your final sketch, and choose Hide Objects under the Display menu.  
  
  

• Select New Sketch under the File menu.  Click on the script window.  The text highlighted in black lets you know what objects are needed to complete the current script.  Create the given objects needed anywhere in your Sketch window.  To play back the script, select the given objects and choose a  playback speed (Step, Play or Fast). Experiment with the different speeds.  Use your script to create a pattern in your sketch window.  
  
  

• There is another way to record a script that is much more convenient -- especially for those of you who do not plan ahead! Go back to your isosceles triangle sketch.  Hide any objects you do not want in your script.  Under the Edit menu, choose Select All.  Under the Work menu, choose Make Script. Voilà! Your script has been recorded based upon your previous construction - mistakes and all!  Save your new script on your disk for future use.  
  
  

• What are the benefits of having scripts?  Think of an example where a script would be helpful to your students.

• Euclid's first proposition in the Elements is, "For a line segment AB, there is an equilateral triangle having the segment as one of its sides".  He provides a proof using a similar construction to the one you have just completed.  Provide your own proof that the triangle you constructed is an equilateral triangle.  
  
  
• Record a script for the construction of a right triangle.  Make sure to drag each vertex to confirm it stays a right triangle.  Be sure to save it for future use on your disk.  Describe how you constructed your right triangle.  You may find the Perpendicular Line command under the Construct menu useful. (Instructor Note: A sketchpad script illustrating the construction of a right triangle has been saved as righttri.gss.)