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?

$d = 0$ 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

Hide solution and development