If radii of two concentric circles are 12 cm and 13 cm, find the length of each chord of one circle which is tangent to the other circle.

Answer

To solve the problem of finding the length of each chord of one circle that is tangent to the other circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii of the Circles**:
- The radius of the smaller circle (Circle 1) is \( r_1 = 12 \) cm.
- The radius of the larger circle (Circle 2) is \( r_2 = 13 \) cm.

2. **Understand the Configuration**:
- The two circles are concentric, meaning they share the same center \( O \).
- A chord of the larger circle (Circle 2) is tangent to the smaller circle (Circle 1).

3. **Draw the Diagram**:
- Let \( A \) and \( B \) be the endpoints of the chord on the larger circle.
- Let \( C \) be the point where the chord \( AB \) is tangent to the smaller circle.
- The center of both circles is point \( O \).

4. **Apply the Right Triangle Concept**:
- Since the chord \( AB \) is tangent to the smaller circle at point \( C \), the radius \( OC \) is perpendicular to the chord \( AB \).
- Therefore, triangle \( OAC \) is a right triangle, where:
- \( OA \) is the hypotenuse (radius of the larger circle) = 13 cm.
- \( OC \) is one leg (radius of the smaller circle) = 12 cm.
- \( AC \) is the other leg (half the length of the chord \( AB \)).

5. **Use the Pythagorean Theorem**:
- According to the Pythagorean theorem:
\[ OA^2 = OC^2 + AC^2 \]
- Substituting the known values:
\[ 13^2 = 12^2 + AC^2 \]
\[ 169 = 144 + AC^2 \]
- Rearranging gives:
\[ AC^2 = 169 - 144 = 25 \]
- Taking the square root:
\[ AC = \sqrt{25} = 5 \text{ cm} \]

6. **Calculate the Length of the Chord \( AB \)**:
- Since \( C \) is the midpoint of \( AB \), we have:
\[ AB = AC + BC = AC + AC = 2 \times AC = 2 \times 5 = 10 \text{ cm} \]

### Final Answer:
The length of each chord \( AB \) of the larger circle that is tangent to the smaller circle is **10 cm**.