Tangent Circles Discussion

by

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.

Make a GSP sketch. Here is how I would do it.

1. Mark of a line segment to represent the radius of
the two original ‘large’ circles

2. Select a point for the center of one circle

3. Construct first circle by center and radius

4. Select a point on the constructed circle.

5. Construct a circle with this point as center and with
the given radius.

6. Construct the small circle.

A. Construct the segment between the two centers

Draw
circle by center an point on the circumference

Measure the area and
circumference of the small circle

Measure the area and
circumference of the large circle

Compare areas:

Compare circumferences:

By algebra,

if
the radius of the small circle
is r

and
the radius of the large circle is R

The situation we have here
is R = 2r.

The ratio of the two areas
will be

The ratio of the two
circumferences will be

In general, the comparison of the two areas is the square
of the ratio of the radii; the comparison of the two circumferences is the
ratio of the radii.

**Extension**:

What if the small circle was tangent to the two given circles as follows? Read the problem carefully – more carefully than it was written. It only says that the small circle is tangent to the other two; it does not say they are tangent at the respective circle centers (as the authors obviously intended from the picture).

**Extension**: What if the two original circles
were of different radii?