Concentric Circles

Concentric circles, a fundamental concept in geometry, can be found everywhere. From the rings of a dartboard to the ripples in the pond, these circles have applications in design, physics, navigation, and more. It's their simplicity and symmetry that make them ideal for theoretical as well as practical understanding. Let’s break down the concentric circles definition, their properties, and more.

1.0What Are Concentric Circles?

So, what are concentric circles?

Concentric circles are rings nested perfectly together. They have the same centre, but different radii. The best example of this is when you toss a pebble into still water. The ripples start spreading out evenly. This is the visual essence of concentric circles.

Take this example. There are three circles: A, B, and C, all with the same centre O. However, each of them has a different radius—2 cm, 3 cm, and 5 cm. Now, even though the radii are different, they all share the same centre. So, in this case, OA ≠ OB ≠ OC. The distance from the centre to each circle’s edge is unequal.

One important property of concentric circles is that they never intersect each other. The rings are perfectly spaced, so they don’t cross paths. The space between any two circles is called an annulus. However, it is important to note that just because one circle is inside the other, it doesn’t mean they are concentric. Circles with different centres don’t count as concentric circles. They are defined by the shared centre, not just their appearance.

2.0Properties of Concentric Circles

Let’s dive into the properties of concentric circles:

• They have the same central point.
• The radius might differ.
• There is no intersection between the circles.
• The difference between the radii gives the distance between the circles.
• Concentric circles have symmetry about the common centre.

3.0Concentric Circle Theorem

In two concentric circles, if a chord of the outer circle touches the inner circle at exactly one point (meaning its tangent to the inner circle), then that chord is bisected at the point of contact.

Let’s dive into how you can represent concentric circles using equations:

Proof of the Concentric Circle Theorem

Let’s break it down with an example.

Given: Two concentric circles, C₁ and C₂, share the same centre O. A chord AB lies in the outer circle C₁ and just touches the inner circle C₂ at point P.

To Prove: AP = PB, i.e., point P bisects the chord AB.

Step-by-Step Proof

Step 1: AB is a chord in the outer circle C₁, and it touches the inner circle C₂ at point P, making it a tangent to C₂.

Step 2: OP is the radius of the inner circle C₂.

Step 3: From geometry, we know: A radius drawn to the point of tangency is perpendicular to the tangent. So, OP ⟂ AB

Step 4: Since OP is perpendicular to chord AB and comes from the centre O, it must bisect AB. That is, AP = PB

Hence proved: The chord AB is bisected at point P.

4.0Solved Problem on Concentric Circles

Problem: Two concentric circles have radii of 4 cm and 9 cm. What is the area of the ring (the shaded region) between them?

Solution:

We calculate the area of the larger circle and subtract the area of the smaller one.

$\text{Area}=\Pi r_2^2-\Pi r_1^2$

$\text{Area}=\Pi (9^2-4^2)=\Pi (81-16)=\Pi (65)$

Area = 3.1416 × 65 = 204.2 cm²

The area of the shaded region between the two concentric circles is 204.2 cm².