Calculating the Length of the Common Chord

Oct 13, 2025 by miodeya.com 147 views

The Intersection of Two Equal Circles: A Mathematical Exploration

In geometry, the intersection of two circles is a fundamental concept that has been studied extensively. When two circles of equal radii intersect such that each passes through the centre of the other, a unique geometric configuration arises. In this article, we will delve into the properties of this configuration and determine the length of the common chord of the intersecting circles.

To begin, let's visualize the scenario. We have two circles, both with radii 'r', intersecting in such a way that each circle passes through the centre of the other. This creates a symmetrical and aesthetically pleasing geometric shape.

Key Properties

Before we proceed to calculate the length of the common chord, let's identify some key properties of this configuration:

The two circles have equal radii 'r'.
Each circle passes through the centre of the other.
The intersection points of the two circles form a symmetrical shape.

Calculating the Length of the Common Chord

To calculate the length of the common chord, we can use the properties of the intersecting circles. Let's consider the diagram below:

  +---------------+
  |               |
  |  Circle 1    |
  |               |
  +---------------+
           |
           |
           v
  +---------------+
  |               |
  |  Circle 2    |
  |               |
  +---------------+

In this diagram, the two circles intersect at points A and B. The common chord is the line segment AB.

Using Geometric Properties

We can use the properties of the intersecting circles to calculate the length of the common chord. Let's consider the right-angled triangle OAB, where O is the centre of both circles.

  +---------------+
  |               |
  |  O           |
  |               |
  +---------------+
           |
           |
           v
  +---------------+
  |               |
  |  A           |
  |               |
  +---------------+
           |
           |
           v
  +---------------+
  |               |
  |  B           |
  |               |
  +---------------+

In triangle OAB, we have:

OA = OB = r (radii of the circles)
∠AOB = 90° (right angle)

Using the Pythagorean theorem, we can calculate the length of AB:

AB² = OA² + OB² AB² = r² + r² AB² = 2r²

Taking the square root of both sides, we get:

AB = √(2r²) AB = √2 * r

In conclusion, the length of the common chord of two circles of equal radii 'r' intersecting such that each passes through the centre of the other is √2 * r.

The correct answer is (b) r√2.

The intersection of two equal circles is a fundamental concept in geometry that has numerous applications in various fields, including physics, engineering, and computer science. Understanding the properties of this configuration can provide valuable insights into the behavior of complex systems and help us develop innovative solutions to real-world problems.

The intersection of two equal circles has numerous real-world applications, including:

Optics: The intersection of two equal circles is used in the design of optical systems, such as telescopes and microscopes.
Engineering: The intersection of two equal circles is used in the design of mechanical systems, such as gears and bearings.
Computer Science: The intersection of two equal circles is used in the development of algorithms for computer graphics and game development.

In conclusion, the intersection of two equal circles is a fundamental concept in geometry that has numerous applications in various fields. Understanding the properties of this configuration can provide valuable insights into the behavior of complex systems and help us develop innovative solutions to real-world problems.
Frequently Asked Questions: The Intersection of Two Equal Circles

In our previous article, we explored the properties of the intersection of two equal circles and calculated the length of the common chord. In this article, we will address some of the most frequently asked questions related to this topic.

Q: What is the intersection of two equal circles?

A: The intersection of two equal circles is a geometric configuration where two circles of equal radii intersect such that each passes through the centre of the other.

Q: What are the key properties of the intersection of two equal circles?

A: The key properties of the intersection of two equal circles include:

The two circles have equal radii 'r'.
Each circle passes through the centre of the other.
The intersection points of the two circles form a symmetrical shape.

Q: How do you calculate the length of the common chord?

A: To calculate the length of the common chord, you can use the properties of the intersecting circles. Let's consider the right-angled triangle OAB, where O is the centre of both circles.

  +---------------+
  |               |
  |  O           |
  |               |
  +---------------+
           |
           |
           v
  +---------------+
  |               |
  |  A           |
  |               |
  +---------------+
           |
           |
           v
  +---------------+
  |               |
  |  B           |
  |               |
  +---------------+

In triangle OAB, we have:

OA = OB = r (radii of the circles)
∠AOB = 90° (right angle)

Using the Pythagorean theorem, we can calculate the length of AB:

AB² = OA² + OB² AB² = r² + r² AB² = 2r²

Taking the square root of both sides, we get:

AB = √(2r²) AB = √2 * r

Q: What are some real-world applications of the intersection of two equal circles?

A: The intersection of two equal circles has numerous real-world applications, including:

Optics: The intersection of two equal circles is used in the design of optical systems, such as telescopes and microscopes.
Engineering: The intersection of two equal circles is used in the design of mechanical systems, such as gears and bearings.
Computer Science: The intersection of two equal circles is used in the development of algorithms for computer graphics and game development.

Q: Can you provide some examples of problems that involve the intersection of two equal circles?

A: Yes, here are some examples of problems that involve the intersection of two equal circles:

Problem 1: Two circles of equal radii intersect such that each passes through the centre of the other. What is the length of the common chord?
Problem 2: A circle is inscribed in a square, and a smaller circle is inscribed in the larger circle. What is the ratio of the areas of the two circles?
Problem 3: Two circles intersect such that each circle passes through the centre of the other. What is the length of the arc intercepted by the common chord?

Q: How do you solve problems involving the intersection of two equal circles?

A: To solve problems involving the intersection of two equal circles, you can use the properties of the intersecting circles and apply geometric concepts, such as the Pythagorean theorem and the properties of right-angled triangles.