Question

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.

Answer 1

Hint: In this question, first we will draw the diagram for the question. To make the diagram we have to use the information given in the question which clearly tells that a right triangle will be formed by joining the two radii and the distance between the centres. Finally, using the property of the chord of a circle, we will get the answer.

Complete step-by-step answer:
In the question, we have:
Radii of two circles are 5 cm and 3 cm respectively.
Distance between the centres of these circles is 4 cm.
These three lengths are forming a right triangle.

So we say that the centre of the circle with radius 3cm must lie on the common chord.

So, above is the diagram for the question, where:
OO’ is the distance between the centres of the two circles.
AO’ = 5cm= Radius of bigger circle.
OA = OB =3cm is the radius of a smaller circle.
AB is the common chord of the circle.
$\because$ Centre O’ of a smaller circle lies on the common chord.
Therefore, we can say that AB is the diameter of the smaller circle.
So, AB = 2 OA = 6 cm.
$\therefore$ Length of the common chord of the circle = 6 cm.

Note: For solving this question, the important thing is to draw the diagram using the information given in the question. Once you make the diagram the question is almost done. You should know the property that the line drawn from the centre on the common chord is the perpendicular bisector of the common chord . This will help you to conclude that the centre of the smaller circle will lie on the common chord in this particular question.