To solve the question provided in the image, we need to calculate the radius of the smaller circle. Here's a breakdown of the question and solution:

Information Given:

1. The diagram shows two circles with centers at E and F, respectively.
2. ABCD is a common tangent to the two circles.
3. The radius of the larger circle (circle with center F) is 15 cm.
4. BC = 22 cm, which is the distance between the centers of the two circles along the tangent line.

To Calculate:

1. Radius of the smaller circle, i.e., the radius of the circle centered at E.

Step-by-Step Solution:

The distance between the two centers (BC = 22 cm) can be represented as the sum of the radius of the larger circle and the radius of the smaller circle: $BC = r_{\text{large}} + r_{\text{small}}$ Substituting the values: $22 = 15 + r_{\text{small}}$

Solving for $r_{\text{small}}$: $r_{\text{small}} = 22 - 15 = 7 \, \text{cm}$

Final Answer:

The radius of the smaller circle is: $\boxed{7 \, \text{cm}}$

Let me know if you want further clarifications or additional details!

Here are five related questions:

1. What is the importance of tangents in solving problems involving circles?
2. How can the Pythagorean theorem be applied to calculate distances involving circles?
3. What other geometric properties could be derived from this diagram?
4. Can you describe other methods to calculate radii or distances between two circles?
5. How can the concept of concentric circles apply in similar problems?

Tip:

Always carefully note the relationship between the radius, tangent, and center-to-center distance when solving problems involving circles.