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Write-up


Title
Tangential Circles

Problem Statement
Two larger circles with equal radii pass through each other's centers. A smaller circle can be created inside the overlapping region so that it is tangent to the other circles. (Tangent means that the circles touch each other but do not cross over each other, nor do they leave any gaps.) Compare the area and circumference of the smaller circle to the area and circumference of the larger circle.

Problem setup

The problem is to create 2 circles of equal diameter both of which pass through the others midpoint. I am then to construct a third circle whose perimeter passes through both of the first two circles midpoints. After construction, I am to investigate what is the relationship between the radii of the circles, the area of the circles, and the circumferences of the circles.

 

Plans to Solve/Investigate the Problem

See above

 

Investigation/Exploration of the Problem

I constructed my first circle in GPS. I the copied the circle and set the second circle perimeter passing through the first circle’s midpoint. This forced the second circle’s midpoint to fall on the first circle’s perimeter. After drawing a line segment between the two midpoints, I constructed a third circle using the line segment as the diameter.

 

At this point I created relationships for the following circle properties:

 

1)      Radii

2)      Area

3)      Circumference

 

Below is my GSP Drawing and calculations:

 

 

As you can see by the calculations, the radii and circumference of the larger circles are twice that of the smaller circle. The area of the larger is 4 times the area of the smaller.

 

Extensions of the Problem

 

The suggested extension stated to discuss the results if the radii of the two large circles are different. If they are different, then the relationship of the two circles change because it becomes impossible for the circles to pass through each other’s radii.

 

It is interesting to note however, as illustrated below, that if a circle is created using the height of a semi circle, that the same relationships between area and circumference exist. Therefore, we know that the circle created using two larger circles diameter’s as tangents and the circle created using a semicircle’s height as the diameter are congruent.

 

Author & Contact
Jim Taylor
jtaylor1@rockdale.k12.ga.us

Link(s) to resources, references, lesson plans, and/or other materials