Two circles touch each other internally. Their radii are 2 cm and 3 cm. The biggest chord of the greater circle which is outside the inner circle is of length

Options

A. `2sqrt2 cm`
B. `3sqrt2 cm`
C. `2sqrt3` cm
D. `4 sqrt 2` cm

Answer

To solve the problem of finding the length of the biggest chord of the greater circle that is outside the inner circle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Circles**:
- We have two circles that touch each other internally.
- The radius of the inner circle (smaller circle) is \( r_1 = 2 \) cm.
- The radius of the outer circle (larger circle) is \( r_2 = 3 \) cm.

2. **Identifying Points**:
- Let \( O \) be the center of the larger circle, and \( K \) be the center of the smaller circle.
- The point where the two circles touch is \( P \).

3. **Finding the Distance Between Centers**:
- Since the circles touch internally, the distance \( OK \) between the centers is given by:
\[ OK = r_2 - r_1 = 3 \text{ cm} - 2 \text{ cm} = 1 \text{ cm} \]

4. **Finding the Length of the Chord**:
- The biggest chord of the larger circle that lies outside the smaller circle is perpendicular to the radius at the point of tangency \( P \).
- Let \( A \) and \( B \) be the endpoints of the chord, and \( C \) be the point of tangency where the smaller circle touches the larger circle.

5. **Using Right Triangle**:
- In triangle \( OAC \):
- \( OA = r_2 = 3 \) cm (radius of the larger circle)
- \( OC = OK + KC = 1 \text{ cm} + 2 \text{ cm} = 3 \text{ cm} \) (since \( KC \) is the radius of the smaller circle)
- By the Pythagorean theorem:
\[ OA^2 = OC^2 + AC^2 \]
Substituting the known values:
\[ 3^2 = 1^2 + AC^2 \]
\[ 9 = 1 + AC^2 \]
\[ AC^2 = 9 - 1 = 8 \]
\[ AC = \sqrt{8} = 2\sqrt{2} \text{ cm} \]

6. **Finding the Length of Chord AB**:
- Since \( AB \) is twice the length of \( AC \):
\[ AB = 2 \times AC = 2 \times 2\sqrt{2} = 4\sqrt{2} \text{ cm} \]

### Final Answer:
The length of the biggest chord of the greater circle which is outside the inner circle is \( 4\sqrt{2} \) cm.