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
