1. Students begin with some exploration using the software. Given a circle and a point
outside the circle, they are asked to conjecture about the relationship between the radius
at the point of tangency and the tangent to the circle. By measuring different angles, they
should find that the radius and the tangent are perpendicular to each other. Try it out for
yourself on Sketchpad .
2. We would then have a discussion of other relationships circles have with right angles.
The students (hopefully) remember that any angle inscibed in a circle whose extended sides pass
through the endpoints of a diameter is a right angle. So, we know we want to construct a right
angle, and we know one way to do that is to have the diameter of a circle. Could it be our
given circle? Students (S): No, because one of the extended sides of our desired angle must
pass through the center. So, we know we need a new circle. We also know that the center of
our given circle will be an endpoint of the diameter. What is the other endpoint? S: The
point where we want the tangent to go through, that is, our given point.
3. Now, we are ready to construct the new circle. Where is the center? S: the midpoint
of the diameter. The students construct this.
4. The intersections of the two circles give us the vertex of the angle we want. So,
that vertex is also the point of tangency. We finish constructing the tangent by connecting
the points.
5. The students can practice the construction using Sketchpad. If they need help, they
could run the script (created by the teacher) over and over to
learn the steps. They should be allowed plenty of time for this practice. You are welcome
to practice , too.
6. Once the students have the construction down, they will do a step-by-step construction
on paper using a compass and straightedge. Since this is a final copy, it should be done neatly.
They need to include written directions for each step, and they need to explain the key that
makes the construction work.
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