We don't know whether the circles are touching each other internally or externally .

so , if the circles are touching each other externally then ,
R1+ R2 = 5.5 + 3.3 = 8.8 cm

or if the circles are touching each other internally then,
R1 - R2 = 5.5 - 3.3 = 2.2 cm
therefore the answer will be both 8.8 cm and 2.2 cm

Theorem: The distance between the centers of two circles that touch each other externally is equal to the sum of their radii.

Given:
- The radius of the first circle is 5.5 cm.
- The radius of the second circle is 3.3 cm.
- The circles touch each other.

To find: The distance between the centers of the circles.

Solution:

Step 1: Identify the centers of the circles
Let O1 and O2 be the centers of the first and second circles, respectively.

Step 2: Draw radii
Draw radii OA1 and OB1 in the first circle, and radii OA2 and OB2 in the second circle. These radii connect the centers O1 and O2 with the points of tangency A1, B1, A2, and B2, respectively.

Step 3: Identify the tangency points
Since the circles touch each other, the radii OA1 and OA2, and OB1 and OB2 are perpendicular to the tangent lines at points A1 and B1, and A2 and B2, respectively.

Step 4: Connect the centers
Draw a line segment connecting the centers O1 and O2. Let this line segment be denoted by C1C2.

Step 5: Draw perpendiculars
From the tangency points A1 and A2, draw perpendiculars to the line segment C1C2. Let the points of intersection be denoted by P1 and P2, respectively.

Step 6: Identify the right triangles
We have right triangles C1P1A1 and C2P2A2.

Step 7: Apply the Pythagorean theorem
In right triangle C1P1A1, we have:
(C1P1)^2 + (P1A1)^2 = (C1A1)^2

In right triangle C2P2A2, we have:
(C2P2)^2 + (P2A2)^2 = (C2A2)^2

Step 8: Substitute the values
Since C1A1 = R1 (radius of the first circle) = 5.5 cm, and C2A2 = R2 (radius of the second circle) = 3.3 cm, we can substitute these values into the equations.

(C1P1)^2 + (P1A1)^2 = (5.5)^2
(C2P2)^2 + (P2A2)^2 = (3.3)^2

Step 9: Solve the equations
We can solve these equations simultaneously to find the values of C1P1 and C2P2.

Step 10: Calculate the distance between the centers
The distance between the centers O1 and O2 is equal to the sum of C1P1 and C2P2.

Final Answer:
The distance between the centers of the two circles is equal to the sum of their radii, which is (5.5 + 3.3) cm = 8.8 cm.