**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**

**Perpendicular Bisector**

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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