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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