Tangent Circle
Kim Seay
EMAT6680
In this exploration, we
are asked to create two circles with equal radii that pass through
each others center. Then, create a smaller circle in the region
where the two larger circles overlap that is tangent to the larger
circles. Now explore the relationship between the circumference
and area of the larger and smaller circle.
I have created one such
example of this below.
I have also created a script which
will allow you to try this for yourself. open a new sketch in
GSP and choose any three points A,B, and C where A is the center
of your original circle and the segment BC will be the radius
of the larger circles. Then click on my script and choose play.
GSP will do the rest of the constructions for you. Click
here to try this.
You can see that whatever three
points you choose, the ratio of the circumferences (from big to
small) will always be 2:1 and the ratio of the areas will be 4:1.
Click here to see an animation of the circles as point
C in the radius moves along the path of another segment(k). This
allows us to see what happens to the ratios as the radius of the
original circle increases and decreases.
Conclusion:
This exploration allows
for the use of many tools in GSP. I think it would work well in
a high school Geometry classroom. Middle school students who have
been taught how to find the area and circumference of a circle
can also benefit from this exploration. My TH grade students would
need a lot of guidance. I would recommend using a pre-made script.
You could allow the students to choose various sets of three points
and form a hypothesis regarding the ratios. I would point out
that they were not actually proving this.
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