Geometry Construction – Fraction and Square Root

 

Motivation:

                        Assume that you have ruler and compass at hand.  If we specify a given length as our unit length and call it as “1”,  how do you use geometry construction approach to create a line segment with length  “2”, “  ”,  “  ” , … ?

 

 

Example:   Given the unit length, construct a line segment with length  .

       Sol:

 

                                   

 

                                    Let   be unit length ( “1” ).  We start from that to create .

1.      From point A,  construct a ray as shown above.

Since we have “1”,  we can find a point P such that

        = 2.   ( can be done by using compass 2 times )

         = 3   ( by using compass 3 times )

 

2.      Construct   .

3.      From point P, construct a line parallel to  and the line crosses

        at one point, denoted as R.

4.      Then   =  .

 

                Proof:

                             1.     //

                                    

                             2.    =  =

 

 

 

Example :  Given the unit length, construct a line segment with length  .

 

       Sol:

                              

 

                          Let   be unit length .

 

1.      From point B, construct a line perpendicular to .

On this perpendicular line, we can find a point C such that

          =   

 

2.      Construct  .  Then   =  .

 

 

             Proof:

                                

                               From Pythagorean theorem,   we have  ² =  ² +  ² = 2

                              So,     =  .

 

 

                  There exist other approaches for similar problems by using tangent line property of a circle, or the altitude of right-angle triangle…  Think of your own method to handle the similar problems.

 

Exercise :  Given the unit length,  construct a line segment with length  .