**Using Classical Geometric Construction Techniques**

**Key Terms**

o Classical geometric construction

o Compass

o Protractor

o Perpendicular bisector

o Angle bisector

**Objectives**

o Recognize several simple tools for use in classical construction and understand how they are used

o Understand how some simple geometric constructions can be performed

For many geometry problems, a rough sketch of the situation is sufficient for solving the problem. Nevertheless, even rough drawings or sketches are sufficient in these cases. In other cases, however, we might want to draw a more accurate diagram with angles, for instance, that are correct in measure. To be sure, computers can be a helpful tool in this regard; various software packages make it possible to construct accurate drawings for construction blueprints, for example. We can, however, construct some amazingly accurate drawings of certain figures using techniques of **classical geometric construction, **which uses simple and readily available tools to draw angles, line segments, and other geometric figures. This article shows you how to perform a few of these constructions.

*Simple Geometric Drawing Tools*

**compass**. A compass is simply a V-shaped device with a needle on one arm and a pencil on the other. Arms of the V are connected such that the angle can be adjusted. Below is a sketch of a compass.

A compass can be used to draw a (nearly) perfect circle. The distance between the needle and the pencil is the radius. The needle is planted in the paper and serves as the pivot point and the center of the circle; simply rotate the compass using the pencil to draw the circle, as shown below.

**protractor**, an example of which is shown below. Protractors show a range of angle measures from 0° to 180°. We will not use protractors much, but it is helpful to be recognize them.

*Basic Geometric Constructions*

We saw above how to construct a circle of a given radius using a compass (simply use a ruler to appropriately position the arms of the compass at the proper distance). Another simple construction is a line segment joining two points. A straightedge (such as a ruler) is ideal for this simple construction, as shown below. Simply align the two points along the straightedge and draw the connecting line with a pencil or pen.

**perpendicular bisector**). Consider the segment below. Somehow, we want to construct the perpendicular bisector, which is shown as a dashed line.

**angle bisector**, which is a line segment that divides an angle into two congruent angles. An angle bisector is shown in the diagram below as a dashed line.

The angle bisector is "halfway" between the two angle segments, and it passes through the vertex of the angle. Thus, we need to find just one point along the angle bisector to allow us to construct the segment. Notice that if we draw a segment perpendicular to the bisector, we have created two congruent triangles (by the ASA condition).

On the basis of these (and other) simple constructions, more complicated constructions can be devised. Although modern mathematics and most engineering and scientific fields do not rely on compasses and straightedges to perform geometric constructions, these are illustrative of some of the principles of geometry and how they can be applied using simple tools.

__Use classical construction techniques to divide the isosceles triangle below into two congruent right triangles.__

__Practice Problem__:

__Let's take a look at what we want to do to divide this triangle.__

__Solution__:

__ Practice Problem:__ Construct a line segment parallel to that shown below.

__ Solution:__ Recall that when parallel lines are cut by a transversal, corresponding angles are congruent. Let's call the segment shown above

*l*. If we draw a line segment perpendicular to

*l*(let's call it

*m*), then we can draw yet another segment

*n*perpendicular to

*m*such that

*n*and

*l*are parallel, as shown below.

*m*.

The new line segment *n* is parallel to the original line segment *l*.

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