Tangent Circles - Assignment 7

April Kennedy

Technology: An Advantage? After working with the exploration, "Given two circles and a point on one of the circles, construct a circle tangent to the two circles with one pont of tangency being the designated point", the benefit of using technology to "see" the investigation became clear. Once the tangent circle is created, the true investigation of where certain loci lie can begin to be discussed

For example, consider the locus of the center of all tangent circles to two given circles. How many constructions would one have to do by pencil and paper to see the emergence of the locus? How long does it take for one to see this path using Geometer's Sketchpad? Not very long! Click here for a demonstration. We are able to animate around the circle and trace the locus of the center. One will see that the path the locus follows is an ellipse with foci at the centers of the given circles.

Using Geometer's Sketchpad, we are able to look at other cases of this same problem. For instance, ,what if the two given circles intersect? How many different circles are there that will be tangent to these two given circles? There are enough that it would take you quite some time to find the locus of the tangent circles on pencil and paper. However, using the animation and trace feature of Geometer's Sketchpad, we can see quite easily that the locus is the hyperbola with foci at the centers of the given circles. This does not prove the statement, but students have an understanding of it and know that it is true. This investigation took a shorter amount of time than if it had been done on pencil and paper and the students are able to investigate even more. Technology: an advantage? I think so!

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