Introduction
Let's go over the basics of circles before diving into the related theorems.
Every point on a
This means that they can be drawn using a
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.
We can prove that by showing how all circles can be matched up using simply
Below drag the circles given. What can we conclude?
One important property of circles is that all circles are
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
points. - A chord is a
line segment whose endpoints lie on theof a circle. - A tangent is a
line that touches a circle at exactlypoint. This is called the point of tangency. - An arc is a section of the
of a circle. - A sector is a part of the
of a circle, bounded by an arc and two radii. - Finally, a segment is a part of the
of a circle, bounded by an arc and a chord.
Tangents
In the below figure we can see a line which is touching the circle at exactly one point. That is the tangent of the circle.
Take the movable point
What do you observe?
Now we see that the line touches the circle at two points. Now move the point
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.
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.
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.