Problem:
Two circles of radii 5cm and 3cm intersect at two points and the distance between their centres is 4cm. Find the length of the common chord.

Solution:

To find the length of the common chord between two intersecting circles, we need to understand the properties of intersecting circles and the relationship between their radii, centers, and common chord.

Step 1: Draw the diagram:
Draw two circles of radii 5cm and 3cm, and mark their centers A and B. The distance between their centers is given as 4cm. Label the points of intersection as C and D.

Step 2: Identify the key points:
In this problem, the key points are:
- Radii: The radii of the circles are 5cm and 3cm.
- Centers: The centers of the circles are A and B.
- Distance between centers: The distance between the centers is 4cm.
- Points of intersection: The points of intersection of the circles are C and D.

Step 3: Apply the relevant properties:
- The line joining the centers of two circles is perpendicular to the common chord at the point of intersection.
- The point of intersection of the common chord and the line joining the centers divides the line into two parts, with lengths equal to the distances from the centers to the points of intersection.

Step 4: Solve the problem:
- Draw a line segment joining the centers A and B. Since the distance between the centers is 4cm, draw a line segment AB of length 4cm.
- Perpendicular bisect the line segment AB and mark the midpoint as M.
- Draw circles with radii 5cm and 3cm centered at A and B, respectively.
- The common chord is CD. Draw the common chord CD such that it intersects the line segment AB at M.
- Since M is the midpoint of AB, AM and BM are both 2cm.
- Using the Pythagorean theorem, we can find the length of CM.
- In the right triangle AMC, AC^2 = AM^2 + CM^2
- 5^2 = 2^2 + CM^2
- CM^2 = 25 - 4
- CM^2 = 21
- CM = sqrt(21)
- Similarly, in the right triangle BMD, BM^2 = BD^2 + DM^2
- 3^2 = 2^2 + DM^2
- DM^2 = 9 - 4
- DM^2 = 5
- DM = sqrt(5)
- Therefore, the length of the common chord CD = CM + DM = sqrt(21) + sqrt(5) cm.

Answer:
The length of the common chord between the two intersecting circles is sqrt(21) + sqrt(5) cm.