# Problems from 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?

1. $$d = 0$$ cm
2. $$d = 9$$ cm
3. $$d = 8$$ cm
4. $$d = 13$$ cm
5. $$d = 12$$ cm
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### 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.

1. Distance between centers is $$0$$, and the radiuses are different, so these are inside concentric circles.
2. Distance between centers is $$9$$ so, as the radius of $$C_1$$ is $$2$$, the circumferences are secant.
3. Distance between centers is $$8$$ so, as the radius of $$C_1$$ is $$2$$, circumferences are internally tangent.
4. $$13$$ cm is greater than the sum of both radiuses, so they are external.
5. $$12$$ cm distance is equal to the sum of the two radiuses, so they are tangent interiors.

### Solution:

1. internally concentric
2. secants
3. interior tangents
4. external
5. internal tangents
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