Relative positions of two circumferences in the plane
Given two circumferences $C_1$ and $C_2$ with given radius $r_1 = 2$ cm and $r_2 = 10$ cm, what is the relative position between $C_1$ and $C_2$ where the radius is given and the distance between the centers $d$ is given?
- $d = 0$ cm
- $d = 9$ cm
- $d = 8$ cm
- $d = 13$ cm
- $d = 12$ cm
Note: to resolve this exercise, it is very fortuitous to take paper and pencil and draw a picture of each case to see the solution clearly.
- Distance between centers is $0$, and the radiuses are different, so these are inside concentric circles.
- Distance between centers is $9$ so, as the radius of $C_1$ is $2$, the circumferences are secant.
- Distance between centers is $8$ so, as the radius of $C_1$ is $2$, circumferences are internally tangent.
- $13$ cm is greater than the sum of both radiuses, so they are external.
- $12$ cm distance is equal to the sum of the two radiuses, so they are tangent interiors.
- internally concentric
- secants
- interior tangents
- external
- internal tangents