Problem:
Two circles of centre A and B of radius 3 and 4 cm respectively intersect at P and B such that AC and BC are tangents to two circle. Find the length of the common chord CD.

Solution:

Let's analyze the given information step by step to find the length of the common chord CD.

Step 1: Draw the diagram
- Draw two circles with centers A and B, and radii 3 cm and 4 cm respectively.
- Mark the points of intersection as P and Q.
- Draw tangents AC and BC to the circles.

Step 2: Find the lengths of AP and BP
- Since AP is a radius of circle A, its length is 3 cm.
- Similarly, BP is a radius of circle B, so its length is 4 cm.

Step 3: Find the lengths of AC and BC
- Since AC is a tangent to circle A, it is perpendicular to AP. Therefore, triangle APC is a right-angled triangle.
- Using the Pythagorean theorem, we can find the length of AC:
AC^2 = AP^2 + PC^2
AC^2 = 3^2 + PC^2
AC^2 = 9 + PC^2

- Similarly, triangle BPC is a right-angled triangle.
- Using the Pythagorean theorem, we can find the length of BC:
BC^2 = BP^2 + PC^2
BC^2 = 4^2 + PC^2
BC^2 = 16 + PC^2

Step 4: Find the length of PC
- Since both AC and BC are tangents to the circles, they are equal in length.
- Therefore, AC = BC.
- Substituting this in the equations from step 3, we have:
AC^2 = 9 + PC^2
BC^2 = 16 + PC^2

- Since AC = BC, we can equate the two equations:
9 + PC^2 = 16 + PC^2
9 = 16

Step 5: Analyze the contradiction
- The equation obtained in step 4 leads to a contradiction (9 = 16), which means that the given information is not consistent.
- Therefore, it is not possible to determine the length of the common chord CD based on the given information.

Conclusion:
- The length of the common chord CD cannot be determined based on the given information, as it leads to a contradiction.