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

The problem involves two circles of radii 5 cm and 3 cm that intersect at two points, with the distance between their centers being 4 cm. We need to determine the length of the common chord formed by their intersection.

Solution

The two circles of radii 5 cm and 3 cm intersect at two distinct points. The line joining the centers of the two circles serves as the perpendicular bisector of the common chord, since the chord is symmetric with respect to this line.

Let the centers of the two circles of radii 5 cm and 3 cm be O and O', respectively. Let the common chord be denoted by PQ. The given parameters are:

• Radius of the first circle, O'P = 5 cm
• Radius of the second circle, OP = 3 cm
• Distance between the centers, OO' = 4 cm

Since the radius of the larger circle (5 cm) is greater than the distance between the centers (4 cm), the smaller circle lies entirely inside the larger circle, but the two circles of radii 5 cm and 3 cm intersect at two points.

Step-by-Step Calculation

1. The common chord, PQ, is bisected perpendicularly by the line OO'.
2. The distance from O to the chord (let's call it the perpendicular distance) can be calculated as the radius of the second circle. Hence, OP = OQ = 3 cm.
3. The length of the chord is the sum of the segments it forms within the smaller circle. Therefore: 
   PQ = OP + OQ = 3 cm + 3 cm = 6 cm.

Final Answer

The length of the common chord between the two circles of radii 5 cm and 3 cm is 6 cm.

This result highlights the geometric relationship between the two circles of radii 5 cm and 3 cm, the distance between their centers, and their intersecting chord.