Linear Constructions with a Compass Help
Introduction to Reproducing and Bisecting Line Segments
The following paragraphs describe how to perform various constructions with line segments. By extension, these same processes apply to rays and lines; you can extend line segments and add arrows as necessary.
Reproducing A Line Segment
Suppose you have a line segment whose end points are P and Q (as shown in Fig. 5-7A), and you want to create another line segment having the same length as PQ . First, construct a “working segment” that is somewhat longer than PQ . Then place a point on this “working segment” and call it R , as shown in drawing B. Next, take the compass and set down the non-marking tip on point P , and adjust the compass spread so the tip of the pencil lands exactly on point Q . By doing this, you have defined the length of line segment PQ using the compass.
Next, place the non-marking tip of the compass down on point R , and create a small arc that intersects your “working segment,” as shown in drawing C. Define the intersection of the “working segment” and the arc as point S . The length of line segment RS is the same as that of PQ , so you have reproduced line segment PQ (drawing D).
Bisecting A Line Segment
Suppose you have a line segment PQ (Fig. 5-8A) and you want to find the point at its center, that is, the point that bisects line segment PQ . First, construct an arc centered at point P . Make the arc roughly half-circular, and set the compass to span somewhat more than half the length of PQ . Then, without altering the setting of the compass, draw an arc centered at point Q , such that its radius is the same as that of the first arc you drew (as shown at B). Name the points at which the two arcs intersect R and S . Construct a line passing through both R and S . Line RS intersects the original line segment PQ at a point T , which bisects line segment PQ (as shown at C).
Constructing Perpendicular Bisectors, Rays Defined at a Point, and Dropping a Perpendicular to a Line
Suppose you have a line segment PQ and you want to construct a line that bisects PQ , and that also passes perpendicularly through PQ . Figure 5-8 shows how this line (called RS in this example) is constructed as a byproduct of the bisection process.
Perpendicular Ray At Defined Point
Figure 5-9 illustrates the construction of a perpendicular ray from a defined point P on a line or line segment.
Begin with the scenario at drawing A. Set the compass for a moderate span, and construct two arcs opposite each other, both centered at point P , that intersect the line or line segment. Call these intersection points Q and R , as shown in drawing B. Increase the span of the compass, roughly doubling it. Construct an arc centered at Q and another arc centered at R , so the two arcs have the same radius and intersect as shown in drawing C. Call this intersection point S . Construct a ray whose initial point is P , and that passes through S . Ray PS is perpendicular to the original line or line segment at the original defined point P .
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