Constructing Tangents
to Circles Using GSP
Topic Index | Geometry Index | Regents Exam Prep Center

 


Version 4.+
   Construct a tangent to a circle from a point not on the circle.
 
1.  Use to draw a circle.  Label it O.  Right clicking on point will allow fast labeling.

2.  Use to locate a point not on the circle.  Label it P.

3.  Select    points O and P by clicking on them.  Use the Segment command under the Construct menu to connect O to P.

4.  Select the segment .  Use the Midpoint command under the Construct menu to bisect the segment.  Label the point M.

5.  Select points O and M.  Use the Segment command under the Construct menu to create a segment from O to M.

6.  Select segment and point M.  Use the Circle By Center+Radius to construct circle M with radius .  Label the intersections with circle O as points R and Q.

7.  Select P and then R.  Use the Ray command under the Construct menu to draw a ray.  Select Q and then R.  Repeat drawing a ray.

 

Does this really work?  Are these really tangents?

Answer:  YES

Draw a segment from O to R and from O to Q forming radii in circle O.  Angles <ORP and <OQP are right angles because an angle inscribed in a semi-circle (in circle M) is a right angle.  The presence of these right angles means that segment is perpendicular to radius in circle O.  The same is true for segment being perpendicular to  in circle O.   Since segments and are perpendicular to radii and at the point of contact on circle O, segments and are tangents to circle O.