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

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Answer

6 CM first show that the line joining centres is perpendicular to the radius of the smaller circle and then the that common chord is the diameter of the smaller circle

Answer

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

Solution:

To find the length of the common chord, we need to use the properties of intersecting circles and apply some geometric principles. Let's break down the solution into several steps.

Step 1: Draw a diagram

To visualize the problem, draw two circles with radii of 5cm and 3cm. Label the centers of the circles as A and B, and mark the intersection points as C and D. Draw the common chord passing through the intersection points.

Step 2: Identify the right triangle

Since the distance between the centers of the circles is 4cm, we can draw a line segment connecting the centers, which we'll label as AB. This line segment forms a right triangle with the common chord as the hypotenuse.

Step 3: Apply the Pythagorean theorem

In the right triangle ABC, with AB as the hypotenuse, we can apply the Pythagorean theorem to find the length of the common chord.

According to the theorem, the square of the hypotenuse (AB) is equal to the sum of the squares of the other two sides (AC and BC).

Using the radii of the circles as the legs of the right triangle, we have:

AC^2 + BC^2 = AB^2

(5cm)^2 + (3cm)^2 = AB^2

25cm^2 + 9cm^2 = AB^2

34cm^2 = AB^2

Step 4: Find the length of AB

Taking the square root of both sides of the equation, we can solve for AB:

AB = √(34cm^2)

AB ≈ 5.83cm

Step 5: Find the length of the common chord

Since the common chord is the hypotenuse of the right triangle, its length is equal to AB.

Therefore, the length of the common chord is approximately 5.83cm.

Conclusion: The length of the common chord between the two circles with radii 5cm and 3cm is approximately 5.83cm.

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