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Chapter 9: Tangents and Secants to a Circle

    Introduction

    Tangents of a Circle

    Number of Tangent to a Circle From Any Point

    Segment of a circle Formed by a Secant

    Summary

    Enhanced Curriculum Support

    Easy Level Worksheet

    Moderate Level Worksheet

    Hard Level Worksheet

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Chapter 9: Tangents and Secants to a Circle > Tangents of a Circle

Tangents of a Circle

In the previous section, we have seen that a tangent to a circle is a line that intersects the circle at only point.

To understand the existence of the tangent to a circle at a point, let us perform the following activities:

Activity 1 : Drag the two point of the red line in different ways and check for the intersection points between the line and the circle. What can be observed?

Only in one position, you will see that it will intersect the circle at only one point (say, P). For all the rest of the cases, there are intersections points (though very close to each other) (or) none at all.

So, we can conclude that there is only tangent at a point of the circle.

In other words:

The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord .

Activity 2 : Connect the points in the figure below. x1- x2, x3-x4 and so on.

When you from the first line to the last you observe that the length of the chord cut by the lines slowly increases and then starts decreasing as you cross the center. In one case, it is zero at the start and in another case, it becomes zero on the other side of the circle. These are the tangents to the circle parallel to the given secant d1 d2.

We can see that there cannot be more than tangents parallel to a given secant.

The common point of the tangent and the circle is called the point of contact and the tangent is said to touch the circle at the point.

Here is a wheel with diameter. Move the circle and observe where it touches the ground.

Do you see any tangent anywhere? In fact, the wheel moves along a line which is a tangent to the circle representing the .

As you drag the wheel along the ground you see that the wheel spokes seem to be at angles whenever they are touching the ground.

This is true for all other points in the circle too. That is, we notice that in all positions, the radius through the point of contact with the ground appears to be at right angles to the tangent. We shall now prove this property of the tangent.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Proof:

Given: Circle with centre O

Line XY- tangent to the circle at a point P

To be proved: OP ⊥ XY.

Construction: Take a point Q on XY other than P and join OQ

Note: The point Q must lie outside the circle, else XY will become a secant and not a tangent to the circle.

Let OQ touch circle at R. Then OQ = +

But OR = .

as OR = OP = Radius of the circle.

OQ = OR + RQ OP.

Since this happens for every point on the line XY except the point P, is the shortest of all the distances of the point O to the points of XY.

So OP ⊥ .

1. By theorem above, we can also conclude that at any point on a circle there can be one and only one tangent.

2. The line containing the radius through the point of contact is also sometimes called the ‘normal’ to the circle at the point.