If radii of two concentric circles are 4 cm and 5 cm, then length of each chord of one circle which is tangent to the other circle, is

Options

A. 3 cm
B. 6 cm
C. 9 cm
D. 1 cm

Explanation

To find 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:**
- Let the radius of the inner circle (C1) be \( r_1 = 4 \) cm.
- Let the radius of the outer circle (C2) be \( r_2 = 5 \) cm.

2. **Understand the Geometry:**
- The chord \( AB \) of circle C2 is tangent to circle C1 at point \( M \).
- The center of the circles is denoted as \( O \).

3. **Draw the Diagram:**
- Draw two concentric circles with center \( O \).
- Mark the tangent point \( M \) on circle C1 where the chord \( AB \) touches it.

4. **Use the Properties of Tangents:**
- Since \( AB \) is tangent to circle C1 at point \( M \), the radius \( OM \) is perpendicular to the chord \( AB \).
- Therefore, \( OM \) is the radius of circle C1, which is \( 4 \) cm.

5. **Apply the Pythagorean Theorem:**
- In triangle \( OMB \), we can apply the Pythagorean theorem:
\[ OB^2 = OM^2 + MB^2 \]
- Here, \( OB \) is the radius of circle C2, which is \( 5 \) cm, \( OM = 4 \) cm, and \( MB \) is the half-length of the chord \( AB \).

6. **Substitute the Known Values:**
- Substitute the known values into the equation:
\[ 5^2 = 4^2 + MB^2 \]
\[ 25 = 16 + MB^2 \]

7. **Solve for \( MB^2 \):**
- Rearranging gives:
\[ MB^2 = 25 - 16 = 9 \]
- Taking the square root:
\[ MB = 3 \text{ cm} \]

8. **Find the Length of Chord \( AB \):**
- Since \( AB \) is twice the length of \( MB \):
\[ AB = 2 \times MB = 2 \times 3 = 6 \text{ cm} \]

### Final Answer:
The length of each chord of circle C2 that is tangent to circle C1 is \( 6 \) cm.

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