Two concentric circles are of radii 13 cm and 5 cm. Find the length of the chord of the larger circle which touches the inner circle.

Answer

To find the length of the chord of the larger circle which touches the inner circle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii**:
- The radius of the larger circle (R) = 13 cm
- The radius of the smaller circle (r) = 5 cm

2. **Draw the Diagram**:
- Draw two concentric circles with the same center O.
- Let A and B be the endpoints of the chord of the larger circle that touches the inner circle at point C.

3. **Understand the Geometry**:
- Since the chord AB touches the inner circle at point C, OC is perpendicular to AB (tangent to the circle).
- Therefore, triangle AOC is a right triangle, where OC is the height from O to the chord AB.

4. **Apply the Pythagorean Theorem**:
- In triangle AOC, we can apply the Pythagorean theorem:
\[ AO^2 = AC^2 + OC^2 \]
- Here, AO = 13 cm (radius of the larger circle), OC = 5 cm (radius of the smaller circle), and AC is half the length of the chord AB.

5. **Substitute the Values**:
- Substitute the known values into the equation:
\[ 13^2 = AC^2 + 5^2 \]
\[ 169 = AC^2 + 25 \]

6. **Solve for AC**:
- Rearranging gives:
\[ AC^2 = 169 - 25 \]
\[ AC^2 = 144 \]
- Taking the square root:
\[ AC = \sqrt{144} = 12 \text{ cm} \]

7. **Find the Length of Chord AB**:
- Since AC is half of AB, we have:
\[ AB = 2 \times AC = 2 \times 12 = 24 \text{ cm} \]

### Final Answer:
The length of the chord AB of the larger circle which touches the inner circle is **24 cm**.