Shape 

Copyright David & Cynthia Thomas, 2009

 

Triangle Recipes--Concepts & Connections

 

Although the directed activity Construct an equilateral triangle using the Geometers Sketchpad approaches the construction of an equilateral triangle using the same logic as Construct an Equilateral Triangle Using Straightedge-and-Compass, it uses a different technology.  The Geometers Sketchpad supports traditional Euclidean straightedge-and-compass constructions, substituting mouse clicks and menu items for paper, pencil, straightedge, and compass.  In addition, The Geometers Sketchpad automates many procedures.  For instance, you may construct perpendicular and parallel lines, segment and angle bisectors, polygon interiors, and many other features (See Figure 1.11).    

 

Figure 1.11: Using The Geometers Sketchpad

 

In the directed activity Construct an equilateral using paper folding, you are asked to fold an equilateral triangle.  One approach to this task is presented in Table 1.3 and Figure 1.12. 

 

Step 1

Beginning with a rectangular sheet of paper with corners ABCD, fold edge BC onto edge AD, dividing the rectangle in half.

Step 2

A second fold contains vertex D and positions point C on the first fold.

Step 3

The third fold is made along C’E and extends to point F. Trim off the excess paper or wrap it around the triangle.

Step 4

Points D, E, and F are the vertices of an equilateral triangle.

Table 1.3: Folding an Equilateral Triangle

 

While the procedure itself is fairly simple, the logic behind the procedure is not.  Consider the following hints as you search for a justification for this construction:

·        What sort of triangle is ΔDC’E?  ΔDC’F?  Why?

·        How is segment FC’ related to segment EC’?  Why?

·        How is ΔDEC’ related to ΔDC’F?  Why?

·        How is ÐC’DF related to ÐC’DE?  Why?

·        How is ÐC’DF related to ÐCDA?  Why?

·        What is the measure of ÐFDE?  Why?

Figure 1.12: Folding an Equilateral Triangle

 

Duplicate a segment of unknown length, Side-Side-Side and Challenge 1.2 focus on general questions related to triangle construction. Every triangle has three sides and three angles.  Using the letter S to indicate a side and the letter A to indicate an angle, one could represent and differentiate all six of these features using the notation S1S2S3 A1A2A3.  This somewhat tedious notation may be simplified to SSSAAA, with the understanding that the three sides and three angles need not be identical, respectively.  Knowing SSSAAA about any given triangle would provide complete information about its most basic features and therefore all the information necessary to construct an exact copy, assuming you know how to copy segments and angles.  A fundamental question in Euclidean geometry asks whether a triangle may be constructed using fewer than all six features.  Posed differently, what information is sufficient to guarantee that a particular triangle may be constructed and/or duplicated?  Figure 1.13 illustrates the basis for two such constructions, commonly called Side-Side-Side and Side-Angle-Side.

Figure 1.13: SSS vs. SAS

 

Table 1.4 and Figure 1.14 present the Side-Side-Side construction in detail.  In this case, the given information is presented in the form of  with known sides AB, AC, and BC (i.e., SSS).  The task is to construct an identical triangle .   Using The Geometers Sketchpad (See 1_SSS.gsp), you may experiment with this construction, taking measurements of sides and angles as necessary. What justification can you supply for the following construction?

 

Step 1

A random point A’ is plotted on a working line.

Step 2

Using A’ as the center and segment AB as the radius, a circle is drawn.  An intersection of this circle and the working line is labeled point B’.

Step 3

Using B’ as the center and segment BC as the radius, a second circle is drawn. 

Step 4

Using A’ as the center and segment AC as the radius, a third circle is drawn.

Step 5

An intersection of circles B’ and A’ is labeled C’. 

Step 6

Draw segments A’C’ and B’C’, creating ΔA’B’C’.. 

Table 1.4: Side-Side-Side Construction

 

Figure 1.14: Side-Side-Side

 

Table 1.5 and Figure 1.15 present the Side-Angle-Side construction in detail.  In this case, the given information is presented in the form of  with known sides AC, and BC and known angle ÐACB.  The task is to construct a duplicate triangle .   Using The Geometers Sketchpad (See 1_SAS.gsp), you may experiment with this construction, taking measurements of sides and angles as necessary. What justification can you supply for this construction?

 

Step 1

A random point C’ is plotted on a working line.

Step 2

Using C’ as the center and segment BC as the radius, a circle is drawn.  An intersection of this circle and the working line is labeled point B’. 

Step 3

Two circles with identical radii are drawn with centers C and C’. 

Step 4

The circle with center C intersects sides BC at point D and AC at point E.  Draw chord DE.

Step 5

The circle with center C’ intersects B’C’ in point D’.  Using D’as the center and chord DE in  as the radius, a circle is drawn that intersects circle C’ at point E’.  A ray is then drawn from C’ through E’, forming Ð B’C’E’.

Step 6

Using point C’ a the center and segment CA in as the radius, a circle is drawn that intersects ray C’E’ at point A’.   

Step 7

A segment is drawn between points A’ and B’, creating .

Table 1.5: Side-Angle-Side Construction

 

Figure 1.15: Side-Angle-Side