Let's explore problem 3a fully, focusing on the learning experience that a student
may have upon first encountering tangent circles.
After constructing the circle in 3a, we could attempt to draw a line through the centers, like so:
Notice that this problem is unique from the exploration problem in Assignment #7
The reason it is different is because all three circles are tangent to each other.
Students should practice with the exploration problem and this problem to
further understand tangent circles.
Let's continue the exploration of this problem to discover relationships.
As you can see, I created a fourth circle, colored purple, with the same radius as the red circle.
Students should recognize that the purple circle is created on the center bisector of
the green and blue circles. Exploring the creation of these circles will give students
valuable experience with GSP, namely using the "Create Circle with Point and Radius"
option and learning to create subsegments.
Let's finish creating the triangle.
As you can see, we have an isoceles triangle just as we did in the exploration problem.
Let's create a triangle formed by a fifth circle, this time using the radius of the green circle.
The new circle and triangle are drawn in black, while the previous circle and triangle are drawn in purple.
Students should (hopefully) recognize that the two triangles may have a relationship.
The two triangles share one side together, but their heights are different because
their heights are the radi of the red and green
circles, respectively.
At this point, I would ask
the students to further explore the triangles.
Are they similar? What would the triangles require in addition to having a shared side
in order to be similar? Can we change anything to make them similar?
These questions can all be derived by these tangent circles, helping students to better
understand not only the properties of circles, but the properties of triangles, as well.