Introduction to Compasss and Straight Edge Instruments
In geometry, a construction is a drawing made with the simplest possible instruments. Constructions are a powerful learning technique, because they force you to think about the properties of geometric objects, independent of numeric lengths and angle measures. Constructions are also challenging intellectual games.
The most common type of geometric construction is done with two instruments, both of which you can purchase at any office supply store. One instrument lets you draw circles, and the other lets you draw straight line segments. Once you have these, you can use them only according to certain “rules of the game.”
Draftsman’s Compass
The draftsman's compass is a device for drawing circles of various sizes. It has two straight shafts joined at one end with a hinge. One shaft ends in a sharp point that does not mark anything, but that can be stuck into a piece of paper as an anchor. The other shaft has brackets in which a pen or pencil is mounted. To draw a circle, press the sharp point down on a piece of paper (with some cardboard underneath to protect the table or desk top), open the hinge to get the desired radius, and draw the circle by rotating the whole assembly at least once around. You can draw arcs by rotating the compass only partway around.
For geometric constructions, the compass must not have an angle measurement scale at its hinge. If it has a scale that indicates angle measures or otherwise quantifies the extent to which it is opened, you must ignore that scale.
Straight Edge
A straight edge is any object that helps you to draw line segments by placing a pen or pencil against the object and running it alongside. A conventional ruler will work for this purpose, but is not the best tool to use because it has a calibrated scale. A better tool is a drafting triangle . Use any edge of the triangle as the straight edge.
Ignore the angles at the apexes of a drafting triangle. Some drafting triangles have two 45° angles and one 90° angle; others have one 30° angle, one 60° angle, and one 90° angle. You aren't allowed to take advantage of these standard angle measures when performing geometric constructions, so it doesn't matter which type of drafting triangle you use.
What's Allowed and Not Allowed When Using a Compass or Straight Edge
What's Allowed
With a compass, you can draw circles or arcs having any radius you want. The center point can be randomly chosen, or you can place the sharp tip of the compass down on a predetermined, existing point and make it the center point of the circle or arc.
You can set a compass to replicate the distance between any two defined points, by setting the non-marking tip down on one point and the marking tip down on the other point, and then holding the compass setting constant.
With the straight edge, you can draw line segments of any length, up to the entire length of the tool. You can draw a random line segment, or choose a specific point through which the line segment passes, or connect any two specific points with a line segment.
What's Not Allowed
Whatever sort of circle or line segment you draw, you are not allowed to measure the radius or the length against a calibrated scale of any sort. You may not measure angles using a calibrated device. You may not make any reference marks on either the compass or the straight edge. (Marking on a straight edge is “cheating,” but referencing a distance using a compass is acceptable, even though the two acts are qualitatively similar!)
Here is a subtle but important restriction: You may not make use of the result of an infinite number of operations (imagining that it is possible), or an infinite number of repetitions of a single operation. That means, for example, that you cannot mentally do a maneuver over and over ad infinitum to geometrically approach a desired result, and then claim that result as a valid construction. The entire operation must be completed in a finite number of steps.
Defining Points and Drawing Line Segments
Defining Points
To define an arbitrary point, all you need to do is draw a little dot on the paper. Alternatively, you can set the non-marking point of the compass down on the paper, in preparation for drawing an arc or circle centered at an arbitrary point. Points can also be defined where two line segments intersect, where an arc or circle intersects a line segment, or where an arc or circle intersects another arc or circle.
Drawing Line Segments
Line segments can be drawn in three ways: arbitrarily, through (or starting at) a single point, and through (or connecting) two points.
When you want to draw an arbitrary line segment, place the straight edge down on the paper and run a pencil along the edge (Fig. 5-1A). You can make it as long or as short as you want, but never longer than the length of the straight edge. If you need to draw a line segment longer than the straight edge, don't align the straight edge with part of the line segment and then try to extend it. Use a longer straight edge, so you can create the entire segment in one swipe.
Fig. 5-1 . At A, construction of an arbitrary line segment. At B, construction of a line segment starting at a single predetermined point. At C, construction of a line segment connecting two predetermined points.
When you want to draw a line segment through a single defined point, place the tip of the pencil on that point (call it point P ), place the straight edge down against the point of the pencil, and then run the pencil back and forth along the edge. If you want the point to be an end point of the line segment, run the pencil away from the point in one direction (Fig. 5-1B).
When you want to draw a line segment through two defined points (call them P and Q ), place the tip of the pencil on one of the points, place the straight edge down against the point of the pencil, rotate the straight edge until it lines up with the other point while still firmly resting against the tip of the pencil, and then run the pencil back and forth along the edge, so the mark passes through both points. If you want the points to be the end points of the line segment, make sure the pencil makes its mark only between the points, and not past them on either side (Fig. 5-1C).
Denoting Rays and Lines
Denoting Rays
In order to denote a ray, first determine or choose the end point of the ray. Then place the tip of the pencil at the end point, and place the straight edge against the tip of the pencil. Orient the straight edge so it runs in the direction you want the ray to go. Move the tip of the pencil away from the point in the direction of the ray, as far as you want without running off the end of the straight edge (Fig. 5-2A). Finally, draw an arrow at the end of the line segment you have drawn, opposite the starting point (Fig. 5-2B). The arrow indicates that the ray extends infinitely in that direction.
Fig. 5-2 . Construction of a ray. First construct a line segment ending at a point (A); then put an arrow at the end opposite the point (B).
Denoting Lines
In order to draw a line, follow the same procedure as you would to draw a line segment. Then place arrows at both ends (Fig. 5-3). A line can be drawn arbitrarily (as shown at A and B), through a single defined point (as shown at C and D), or through two defined points (as shown at E and F).
Fig. 5-3 . At A and B, construction of an arbitrary line. At C and D, construction of a line through a single predetermined point. At E and F, construction of a line through two predetermined points.
Drawing Circles and Arcs
Drawing Circles
To draw a circle around a random point, place the non-marking tip of the compass down on the paper, set the compass to the desired radius, and rotate the instrument through a full circle (Fig. 5-4A). If the center point is predetermined (marked by a dot), place the non-marking tip down on the dot and rotate the instrument through a full circle.
Fig. 5-4 . At A, a compass is used to draw a circle around a randomly chosen center point. At B, a compass is used to draw an arc centered at a predetermined point.
Drawing Arcs
To draw an arc centered at a random point, place the non-marking tip of the compass down on the paper, set the compass to the desired radius, and rotate the instrument through the desired arc. If the center point is predetermined (marked by a dot), place the non-marking tip down on the dot and rotate the instrument through the desired arc (Fig. 5-4B).
Compass and Straight Edge Practice Problems
PROBLEM 1
Define a point by drawing a dot. Then, with the compass, draw a small circle centered on the dot. Now construct a second circle, concentric with the first one, but having twice the radius.
SOLUTION 1
Figure 5-5 illustrates the procedure. In drawing A, the circle is constructed with the compass, centered at the initial point (called point P ). In drawing B, a line segment L is drawn using the straight edge, with one end at point P and passing through the circle at a point Q . The line segment extends outside the circle for a distance considerably greater than the circle's radius. In drawing C, a circle is constructed, centered at point Q and leaving the compass set for the same radius as it was when the original circle was drawn. This new circle intersects L at point P (the center of the original circle) and also at a new point R . Next, the non-marking tip of the compass is placed back at point P , and the compass is opened up so the pencil tip falls exactly on point R . Finally, as shown in drawing D, a new circle is drawn with its center at point P , with a radius equal to the length of line segment PR .
Fig. 5-5 . Illustration for Problem 1.
PROBLEM 2 Draw three points on a piece of paper, placed so they do not all lie along the same line. Label the points P, Q , and R . Construct Δ PQR connecting these three points. Draw a circle whose radius is equal to the length of side PQ , but that is centered at point R .
SOLUTION 2
The process is shown in Fig. 5-6. In drawing A, the three points are put down and labeled. In drawing B, the points are connected to form Δ PQR . Drawing C shows how the non-marking tip of the compass is placed at point Q , and the tip of the pencil is placed on point P . (You don't have to draw the arc, but it is included in this illustration for emphasis.) With the compass thus set so it defines the length of line segment PQ , the non-marking tip of the compass is placed on point R . Finally, as shown in drawing D the circle is constructed.
Fig. 5-6 . Illustration for Solution 2
PROBLEM 3
Can the non-marking tip of the compass be placed at point P , and the pencil tip placed to draw an arc through point Q , in order to define the length of line segment PQ in Problem 2?
SOLUTION 3
Yes. This will work just as well.
Practice problems for these concepts can be found at: Compass And Straight Edge Practice Test.
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