Geometry Construction – Tangent Line of a Circle

 

 

Question:  Given a circle and a point outside the circle,  construct the tangent line(s) passing the point.

 

                     

        Sol:

 

                           Given a circle O and a point P outside the circle, 

                            We will construct the tangent lines passing P.

 

1.      Construct . 

      Find the midpoint of  and denote it as point S.

2.  Use point S as the center and  as diameter to draw a circle.

                                 The circle crosses circle O at two points, denoted as Q , R.

 

3.      Construct  and .

Then the two lines are tangent lines of circle O and they pass P.

 

          Proof:

                            1.    is the diameter of circle S

                                     PQO = 90⁰ .

                                 And   is the radius of circle O

                                       is a tangent line.

 

2.      Using similar approach can prove  is also a tangent.

 

 

 

                  The technique for constructing tangent lines of a circle is quite useful.  We introduce two questions below to illustrate how to use it.

 

 

 

 

 

 

 

Question :  Give a line and two points sitting on the same side of the line as shown:

 

                             

                    Let R be a sliding point on L.  Find the point S on L such that

                                          +     +   

 

                   Sol:

1.      From P, construct a perpendicular line to L and

that line crosses L at a point, denoted as N.

2.      On ,  find the point T such that

         =   , as shown in the diagram .

3.      Construct   and it crosses L at a point, denoted as S.

Then S is the point we are looking for.

 

                   Proof:

                                1.     L ,  =

                                        = ,  =

                                2.  In  TQR,    +  >  =  +

                                3. From 1 and 2, we have

                                           +   >  +   as long as  S  T .

 

 

 

 

 

 

Question :  Two points P and Q are on the same side of a line L.

                   Try to find a point R on L such that

                               

                                    2 = 2  1

 

                        

                                                  

 

 

        Sol:

1.      From point Q, construct a perpendicular line to L .

The line crosses L at one point, denoted as N.

And we specify a point on this perpendicular line such that

  =    as shown in the diagram.

 

2.      From P, construct a line perpendicular to L.

Use P as center and the distance from P to L as radius to construct a circle.

 

3.      From point S,  construct the tangent line to the circle P.

And the tangent line crosses L at a point, denoted as R.

Then R is the point we are looking for.

 

       Proof:

 

                        1.    L 

                              L is also a tangent line of the circle P.

                        2.  PRU   PRV  ( RHS congruence theorem, or Pythagorean theorem )

                               PRU =  PRV

                        3.   L is the perpendicular bisector of 

                                 QRN   SRN

                                 QRN =  SRN ( =  URV )

4.      From 2 and 3,   QRN = 2  PRU .