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# Linear Constructions with a Compass Help

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

## 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).

Fig. 5-7 . Reproducing a line segment.

### 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).

Fig. 5-8 . Bisecting a line segment, and constructing a perpendicular bisector.

## 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.

Fig. 5-8 . Bisecting a line segment, and constructing a perpendicular bisector.

### 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 .

Fig. 5-9 . Constructing a ray perpendicular to a line or line segment.

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