Using Sketchpad, construct a circle, a point on it, and a line through this point. Drag the line around, until it appears to become tangent to the circle. Is there a special relationship between the line and the circle? Now draw the segment joining the center of the circle and the point on it. What angle do the two lines seem to form? Measure it using the "measure angle" feature. Is this angle special for the tangent line?
How can we use the previous property to construct a tangent line to a circle? Construct a circle, and a point on it, and then construct the tangent line at that point.
This is all good, but suppose we are given a circle centered at O, and a point Q outside the circle. There are two lines through Q which are tangent to the circle, and we want to be able to construct (one of) them. If we could find which point P on the circle is the point of tangency, then we could construct the line as before. How do we find P? Look at what we know so far: the tangent line should be perpendicular to the radius OP, and it should pass through the given external point Q. Let's look at all the lines satisfying these conditions.
So, construct a circle, and a point Q not on it. Now construct a line through the center O of the circle to some point A, and construct a perpendicular line to it through Q. Let P be the point of intersection of these two perpendicular lines. Now drag point A around, moving the whole configuration, and trace point P. What figure do you seem to get? Make a conjecture. Construct the figure, and check whether your conjecture was right or not.
Of all these possible positions of the points P, which makes line PQ tangent to the circle? Is there more than one? How would you find it (them?) using the tools available in Sketchpad? Putting this all together, given a circle and an exterior point Q, how do you construct the line through Q tangent to the circle? Create a Sketchpad script of this procedure that can then be used again and again, with any circle and exterior point. You may want to hide the figures used in the construction so that only the tangent line appears.
Author: Eduardo Tabacman, revised and edited by Evelyn Sander
Comments to: email@example.com
Created: Jun 09 1996 --- Last modified: Jun 11 1996