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

See development and solution

### Development:

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.

### Solution:

- internally concentric
- secants
- interior tangents
- external
- internal tangents