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Circles

    Introduction

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10th class > Circles > Introduction

Introduction

Let's go over the basics of circles before diving into the related theorems.

Every point on a circle has the same distance from its center. This means that they can be drawn using a compass:

There are three important measurements related to circles that you need to know:

  • The radius is the distance from the center of a circle to its outer rim.
  • The diameter is the distance between two opposite points on a circle. It goes through its center, and its length is the radius.
  • The circumference (or perimeter) is the distance around a circle.
  • The chord is a line segment joining any two points on the circumference of the circle.

One important property of circles is that all circles are similar. We can prove that by showing how all circles can be matched up using simply translations and dilations:

Apart from center, radius, diameter and circumference , there are many geometric elements related to a circle, which we’ll need to solve more complex problems:

  • A secant is a line that intersects a circle at two points.
  • A chord is a line segment whose endpoints lie on the circumference of a circle.
  • A tangent is a line that touches a circle at exactly one point. This is called the point of tangency.
  • An arc is a section of the circumference of a circle.
  • A sector is a part of the interior of a circle, bounded by an arc and two radii.
  • Finally, a segment is a part of the interior of a circle, bounded by an arc and a chord.

Tangents

In the above figure we can see a line which is touching the circle at exactly one point. That is the tangent of the circle. Now move the point b1 to x1.

Now we see that the line touches the circle at two points. Now move the point b1 to x2.

In this scenario we see that the line does not touch the circle even once. These are the only possible combinations when we consider a line and a circle.

We can deduce that when it comes to listing the possible combinations for a circle and line, we get three case:

(i) Circle and line don't intersect at all.

(ii) Circle and line intersect at one point (giving a tangent).

(iii) Circle and line intersect at two points.

Pulley

You might have seen a pulley fitted over a well which is used in taking out water from the well. Here the rope on both sides of the pulley, if considered as a ray, is like a tangent to the circle representing the pulley.

Is there any position of the line with respect to the circle other than the types given above?

You can see that there cannot be any other type of position of the line with respect to the circle. In this chapter, we will study about the existence of the tangents to a circle and also study some of their properties.