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

Answer

To solve the problem, we need to find the length of the biggest chord of the greater circle that is outside the inner circle. Here are the steps to arrive at the solution:

### Step-by-Step Solution:

1. **Identify the Radii of the Circles:**
- Let the radius of the smaller circle (inner circle) be \( r = 2 \) cm.
- Let the radius of the greater circle be \( R = 3 \) cm.

2. **Understand the Configuration:**
- The two circles touch each other internally at point \( P \).
- The center of the smaller circle is \( O \) and the center of the greater circle is \( Q \).

3. **Calculate the Distance Between the Centers:**
- The distance \( OP \) (from the center of the smaller circle to the point of contact) is equal to the radius of the smaller circle, which is \( 2 \) cm.
- The distance \( QP \) (from the center of the greater circle to the point of contact) is equal to the radius of the greater circle, which is \( 3 \) cm.
- Therefore, the distance \( OQ \) between the centers of the circles is:
\[ OQ = OP + PQ = 2 + 3 = 5 \text{ cm} \]

4. **Determine the Length of the Chord:**
- The largest chord of the greater circle that is outside the inner circle is perpendicular to the line segment \( OQ \) and bisects it at point \( K \).
- The distance \( KQ \) from the center \( Q \) to point \( K \) is:
\[ KQ = QP - OP = 3 - 2 = 1 \text{ cm} \]

5. **Apply the Pythagorean Theorem:**
- In triangle \( AKQ \), where \( A \) is a point on the chord, we have:
- \( AQ = R = 3 \) cm (the radius of the greater circle),
- \( KQ = 1 \) cm (the distance from the center to the midpoint of the chord),
- Let \( AK = x \) cm (half the length of the chord).
- According to the Pythagorean theorem:
\[ AQ^2 = AK^2 + KQ^2 \]
Substituting the known values:
\[ 3^2 = x^2 + 1^2 \]
\[ 9 = x^2 + 1 \]
\[ x^2 = 9 - 1 = 8 \]
\[ x = \sqrt{8} = 2\sqrt{2} \text{ cm} \]

6. **Calculate the Length of the Chord \( AB \):**
- Since \( AB = 2 \times AK \):
\[ AB = 2 \times 2\sqrt{2} = 4\sqrt{2} \text{ cm} \]

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