Olive & Oppong: Transforming Mathematics with GSP 4, page 11 Chapter 1 Euclid’s Construction Tools Euclid was a Greek mathematician who lived about three centuries before the common era (c. 300 BC). His geometric treatise, referred to as Euclid’s Elements, has been the most enduring and widely used mathematical work in the history of mathematics. Euclid lays forth an axiomatic system in the Elements whereby most of the theorems of plane geometry can be derived through logical deduction from a small set of definitions, postulates and axioms (statements that are accepted without proof, such as “things equal to the same thing are equal to one another”). Euclid’s Elements also contains many geometric construction problems that had to be carried out using very specific tools: a collapsible compass and a straightedge. Euclid’s collapsible compass could be used to draw a circle or an arc of a circle from a given point (as center of the circle) passing through another given point (that determines the radius of the circle or arc). Because the compass was collapsible, however, it could not be used to transfer a given distance between two points onto a line that did not contain those two points (as we can do with a modern compass whose radius can be fixed and then the compass moved to another point as center for duplicating the radius on a new line). Euclid’s straightedge was also different from a modern ruler in that it had no metric. It could only be used for constructing line segments or lines passing through given points. It could not be used for measuring distances between points or for duplicating a given distance (by making marks on the straightedge). Because of these restrictions, even the most basic constructions were fairly complex. For instance, start with an arbitrary line segment and try to duplicate that line segment on a line not containing the line segment, as in Figure 1.1. Olive & Oppong: Transforming Mathematics with GSP 4, page 12 B A C D Figure 1.1: Duplicate Segment AB on Line CD Following the strict rules of Euclidean construction, this seemingly simple task requires the construction of 12 arcs (or circles) and 6 extra lines (see Figure 1.3). The rules are as follows: 1. You may draw a line or line segment between any two points. 2. From any point you may draw an arc or circle of arbitrary radius. 3. From any point you may draw an arc or circle that passes through any other point. 4. From any point you may draw an arc or circle to intersect any line or another arc or circle that has a different center. 5. You may construct points at the intersections of any lines, line segments, circles or arcs. Given these strict rules, one approach to duplicating the segment AB on the line CD at point C, is to construct a parallelogram with sides CA and AB. However, in order to construct a line through a point parallel to a given line or segment (following the above rules) requires the construction of pairs of perpendicular lines (why?). It turns out that the construction of a line perpendicular to a given line through a given point is the basic building block for most of the possible constructions using Euclidean construction tools. Figure 1.2 illustrates the construction of a line perpendicular to line AB through the point C (not on AB). Olive & Oppong: Transforming Mathematics with GSP 4, page 13 G c2 c3 B F c1 E A C H Figure 1.2: Construction of a Perpendicular to a Line Using Euclidean Construction Tools The steps in the construction of the perpendicular to line AB through C are as follows: 1. Draw a circle (c1), center C to intersect line AB in two points E and F. 2. Draw a circle (c2), center E passing through point F. 3. Draw a circle (c3), center F passing through point E. 4. Draw a line through the intersection points, G and H of circles c2 and c3. This line passes through point C and is perpendicular to line AB. Activity 1.1: Define “perpendicular lines” in your own words. Provide a rationale for why the four construction steps above produce a line perpendicular to line AB through point C. Using similar steps, construct the line through C that is perpendicular to the line GH. Will this line be parallel to line AB? If so, why? Olive & Oppong: Transforming Mathematics with GSP 4, page 14 B A V C X D Figure 1.3:Duplicating Segment AB on Line CD at Point C Using Euclidean Construction Tools. Figure 1.3 illustrates the complexity of the construction steps needed to duplicate a segment following the Euclidean construction rules. Segment AB is congruent to segment CX on line CD. The parallelogram ABVC was constructed through several applications of the steps needed to construct a line parallel to a given line (or segment) through a point not on that line (see assignment 1.1 above). Twelve construction circles and 6 construction lines were needed to construct segment CX on line CD. If we were to use a “modern” (non-collapsible) compass, we could simply set the radius of our compass, using segment AB, and with the point (center) of the compass at point C, draw an arc (or circle) with radius AB to intersect line CD at point X (basically two simple steps). Most (if not all) high-school geometry texts used in our schools today allow the use of a modern compass in construction problems. The ancient Greeks also used their physical compasses to mark off distances (as with a modern, non-collapsible compass). Olive & Oppong: Transforming Mathematics with GSP 4, page 15 Euclid’s rules of construction were mathematical rules that avoided the necessary error that could creep into a construction when using the compass to duplicate a length. In practice, of course, error could be introduced when drawing arcs or circles with a compass (the radius may not stay fixed during the drawing process). Euclid’s constructions were geometric exercises to train the mind rather than engineering problems and so the mathematical purity was what mattered. In the following construction problems you may use a modern (non-collapsible) compass to duplicate lengths of segments. For each construction problem, first write a definition in your own words for the object that you are to construct, then carry out the construction using only a non-collapsible compass and straight edge. Activity 1.2: Construction Problems 1. Construct the perpendicular bisector of a given segment. 2. Construct the angle bisector of a given angle. 3. Construct a line parallel to a given line through a given point not on the line. 4. Construct the midpoint of a given segment. 5. From a point not on a given line, construct the shortest segment from the point to the line. 6. Construct an equilateral triangle given one side. 7. Duplicate a given angle on a line or ray that is not part of the given angle. For each of the above constructions, write as many properties as you can find concerning the constructed objects. The Geometer’s Sketchpad Construction Tools The free-hand tools that Sketchpad provides can be regarded as electronic versions of Euclid’s construction tools. Sketchpad provides a Point tool for creating free points or points at the intersection of two objects; a Straightedge tool constructing line segments, rays or lines; and a Circle tool for for creating circles based on two points (center and radius point). The Circle tool really does operate as a collapsible compass as there is no way to fix the radius of the circle other than by designating the two points. The tools are easy to use. Simply choose one of the three tool buttons by a mouse click and then click (either on an empty part of the screen or on a given point or Olive & Oppong: Transforming Mathematics with GSP 4, page 16 object) to establish the initial point, and then drag the mouse to a different position and click to create the terminal point of the object you are creating. When using the Point tool, of course, only one click is needed to create a point in the sketch. The Straightedge tool is really three tools in one. As shown in the icons above it is a Segment tool. In order to change it to either a Ray or Line tool, simply hold the mouse button down when selecting the tool until the button expands to the right showing all three icons. Then simply slide over to the desired icon and release the mouse button. This Straightedge tool will remain the tool now shown until changed by selecting one of the other icons in the same manner. The following icon is the Line tool: and this is the Ray tool: . Experiment with each of the free-hand tools. Create circles, lines, rays and segments in your sketch. Place points on lines and on circles. Create a circle beginning and ending on points that are already in your sketch. The tool with the letter A is the Labeling tool. Choose this tool and then click on any of the points or other objects that you have created in your sketch. Labels should appear next to the object. Clicking on the object again with the Label tool should make the label disappear. Start with a new sketch. Create a circle with center labeled A and radius point labeled B. Now create a second circle starting with the terminal point of your first circle (B) and ending on the initial point of your first circle, point A (see Figure 1.4). A B Olive & Oppong: Transforming Mathematics with GSP 4, page 17 Figure 1.4: Two Circles Sharing the Same Points but in Reverse Order. Activity 1.3: Using the construction in figure 1.4 as a building block, carry out the construction problems 1-6 above using only Sketchpad’s free-hand tools. Why would it be very difficult to duplicate an angle (construction problem 7) just using these free-hand tools? Sketchpad provides a way of making constructions with the electronic equivalent of a modern (non-collapsible) compass. The Construct menu contains short cuts for construction problems 1-5 above and also provides the option of constructing a circle given a center point and a radius (a selected segment). This construction option (Circle by Center + Radius) provides the shortcut for duplicating segments as with a modern compass. With the addition of this construction menu option, duplicate an angle in Sketchpad. [Note: To use the Construction menu options you must first select the necessary starting objects (and nothing else). Thus, to use the Circle by Center + Radius option you must have one point and one segment only selected in your sketch.] Assignment 1.1 Write a brief report explaining how the Sketchpad freehand tools are electronic equivalents of Euclid’s construction tools. In the next section we shall be using Sketchpad’s Construct menu options as shortcuts for the basic constructions and our exploration of the geometry of triangles. Investigating triangle congruencies and special triangles Activity 1.4 Use the construction tools and menu items in GSP to duplicate a triangle based on the following given attributes of the triangle: Two sides and the included angle (SAS) Three sides (SSS) Two angles and the included side (ASA) Two sides and a non-included angle (SSA) Does this last one give you a unique triangle? Reflection When duplicating objects in GSP you can test your duplicate by changing the original and checking to see that your duplicate changes accordingly. Does this dynamic test help or hinder your understanding of the construction steps you used? Does it provide feedback for correcting a construction that does not behave according to the original shape?