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To find the length of \(DC\) using the Pythagorean theorem, we'll follow these steps:

### Given Data:
- The radius of pulley \(A\) is \(r_A = 3\) cm.
- The radius of pulley \(B\) is \(r_B = 8\) cm.
- The distance between the centers \(A\) and \(B\) is \(AB = 13\) cm.

### Problem Analysis:
We are asked to find the length of \(DC\), which is the distance between the point \(D\) on the smaller pulley and the point \(C\) on the larger pulley.

### Step 1: Determine the Geometry
Points \(D\) and \(C\) are where the common external tangents to both circles touch the circles. The line \(DC\) is tangent to both circles.

### Step 2: Use the Pythagorean Theorem
Let's denote the distance between the points where the tangents touch as \(DC = L\).

- The distance between the centers \(A\) and \(B\) is \(AB = 13\) cm.
- The difference in radii is \(r_B - r_A = 8 - 3 = 5\) cm.

Since \(AB\) is the hypotenuse, and \(DC\) and \(r_B - r_A\) form the other two sides of the right triangle, by the Pythagorean theorem:

\[
AB^2 = (r_B - r_A)^2 + L^2
\]

Substituting the values:

\[
13^2 = 5^2 + L^2
\]

\[
169 = 25 + L^2
\]

\[
L^2 = 169 - 25
\]

\[
L^2 = 144
\]

\[
L = \sqrt{144} = 12 \text{ cm}
\]

### Final Answer:
The length of \(DC\) is \(12\) cm.

Would you like further details or have any questions?

### Related Questions:
1. How do you find the length of the internal tangent between two circles?
2. What is the relationship between the radii and the distance of the tangents?
3. How can this principle be applied in a real-world pulley system?
4. Can you explain why the Pythagorean theorem is applicable in this situation?
5. How would the problem change if the radii of the pulleys were different?

### Tip:
When dealing with problems involving circles, tangents, and distances, always consider whether the Pythagorean theorem can simplify your calculations by recognizing right triangles within the problem.

Learn how to find the length of DC using the Pythagorean theorem in circle geometry. This problem involves calculating distances between points on circles, applying the Pythagorean theorem, and understanding circle geometry principles. Suitable for Grades 9-12.

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