Basic Geometry Construction with Ruler and Compass

 

                     Ancient people started with ruler and compass as the tools for geometry construction. However, there are some rules associated with using ruler and compass:

 

1.      Ruler is used as “straight edge”.  In other words, there is no mark on it. It is only used to draw lines.

2.      Compass is used to snap the length as radius to draw circle.

3.      The number of steps must be finite.

 

             With ruler and compass along with the rules,  we say that it is “Geometry Construction”.    Here, we introduce the steps for the construction of some figures without proof.   Once the theorems for proof are established, the proof for showing that those steps are correct are trivial tasks.  That will be done later.

 

 

 

Example 1 ( Triangle ):  

Given three line segments, use the lengths of the three line segments to construct a triangle.

 

                           

 

1.      Draw a line using ruler .

2.      On this line, let’s choose a point randomly and denote it as point A.

3.      Use compass with the first line segment as radius and A as center to draw arc.  We denote the intersection of the arc and the line as point B.

4.      Use compass with the second line segment as radius and A as center to draw an arc.

5.      Use compass with the third line segment as radius and point B as center to draw arc; let’s denote the intersection of the arc and the previous arc in step 4 as point C.

6.      Construct   and .  Then

        ABC is the triangle we want.

 

 

 

 

 

 

 

Example 2 ( Angle )

                 Given   A,  construct an angle with the same magnitude as it.

                   

 

1.      Construct   using ruler.

2.      Use compass to draw an arc with proper length and A as center. The arc crosses the two sides of the angle at two points as shown in the diagram. The two points are denoted as point C and point D.

 

Use the same length as radius and point B as center to draw arc. The arc crosses  at one point as shown.  Let’s denote it as point E.

 

3.      Use E as center and  as radius to draw an arc.

Let’s denote the intersection of the arc and the previous arc as point F.

 

4.      Construct  .  Then  FBE  is what we want.

 

 

Example 3 ( Midpoint of Line Segment )

                   Given ,  find the midpoint of .

 

                                             

 

 

1.      Use A as center and length ( larger than  ) to draw an arc .

2.      Use B as center and the same length to draw an arc; the arc crosses the previous arc in step 1 at points P and Q.

3.      Construct . Let the intersection of   and  be denoted as M.

Then M is the midpoint of .

 

 

 

 

 

 

Example 4 ( Perpendicular Bisector of  Line Segment )

                   A bisector of a line segment is a line passing the midpoint of the line segment.  Perpendicular bisector of a line segment is the bisector that is perpendicular to the line segment.

                  

                 To construct the perpendicular bisector, just follow the steps we use for finding the midpoint in previous example.    is the perpendicular bisector of .

 

 

 

 

 

Example 5 ( Perpendicular Line )

                Given a line segment and a point on it, find the line crossing that point and perpendicular to the line segment.

 

                   

 

                           Let L be a line and A be a point on L. 

                          We would like find a line passing A and perpendicular to A.

1.      Use point A as the center to draw a circle.

The circle crosses L at two points. Denote them as B and C.

By doing so, A is the midpoint of   .

 

2.      Use the steps shown in the previous example to construct perpendicular bisector of   .

Then the perpendicular bisector is what we are looking for.

 

 

 

 

 

 

 

 

Example 6 ( Perpendicular Line from a Point outside a Line )

                 Give a line L and a point P not on L, find the line passing P such that the line is perpendicular to L.

 

                 

 

1.      Use P as center and length larger enough as radius to draw a circle such that the circle crosses L at two pints, denoted as A and B.

2.      Construct the perpendicular bisector of  . Then the line is what we are looking for.

 

 

Example 7 ( Parallel Lines )

                        Given a line L and a point A not on L,  find a line passing A and the line is parallel to L.

 

                    

1.      Construct a line M such that M passes A and M  L.

2.      Construct a line N such that N passes A and N  M.

Then N is the line we are looking for.