Circles

10. Circles

10.1 Introduction

In geometry, a circle is defined as the set of all points in a plane that are equidistant from a central point, known as the centre. The consistent distance from the centre to any point on the circle is called the radius.

Key terminology related to circles includes:

• Chord: A line segment with both endpoints on the circle.
• Segment: A region bounded by a chord and the arc that it subtends.
• Sector: A 'pizza slice' shaped area defined by two radii and the arc between them.
• Arc: A portion of the circle's circumference.

The interaction between a circle and a line can result in several configurations:

1. The line may not touch the circle at all (non-intersecting line).
2. The line may cut the circle creating two points of intersection, qualifying it as a secant.
3. The line may touch the circle at exactly one point, qualifying it as a tangent.

10.2 Tangent to a Circle

A tangent is defined as a line that intersects the circle at a single point. The chapter provides activities to demonstrate how a tangent exists at a point on the circle by rotating a wire (representing the line) and observing the single intersection point.

When gradually approaching the tangent position, the definition solidifies that a tangent can also be seen as a special case of a secant, where both endpoints of the secant converge at one point (the point of contact).

Properties of Tangents:

• The tangent to a circle at any point is always perpendicular to the radius drawn to the point of contact.
• This property is formally established in Theorem 10.1, which asserts that the line formed by the radius to the point of contact is the shortest distance from the center to the tangent line.

Example: Consider a wheel rolling along the ground; the line touching the circle at the bottom of the wheel represents the tangent, and the radius to this point is perpendicular to this tangent.

10.3 Number of Tangents from a Point on a Circle

Three scenarios regarding the number of tangents from a point relative to the circle can be established:

1. Inside the Circle: No tangents can be drawn.
2. On the Circle: Only one tangent can be drawn.
3. Outside the Circle: Exactly two tangents can be drawn.

This section emphasizes that the lengths of the two tangents drawn from an external point to a circle are equal, proven in Theorem 10.2 by analyzing the right angle triangles formed by the radius and the tangents.

10.4 Summary

The chapter concludes with a succinct summary of essential points related to tangents:

1. A tangent to a circle touches it at exactly one point.
2. The tangent line is always perpendicular to the radius at the point of contact.
3. Tangents drawn from a point outside the circle are equal in length.

Key Points

1. A circle is made up of points equidistant from a center.
2. Tangents intersect at only one point of the circle.
3. Secants intersect the circle at two points.
4. A line can be non-intersecting with respect to a circle.
5. The tangent is perpendicular to the radius at the point of contact.
6. A tangent is a secant when both endpoints coincide.
7. From a point inside a circle, there are no tangents; from a point on it, there is one tangent; from a point outside, there are two tangents.
8. Lengths of tangents from an external point are equal.
9. The normal line at the contact point is also the radius through that point.
10. The chapter shows various proofs and examples reinforcing tangential concepts.

1. A circle is defined by points at a constant distance from a centre.
2. Tangents meet circles at only one point.
3. Secants intersect circles at two points.
4. Tangents are always perpendicular to the radius.
5. A point inside a circle has no tangents; a point on it has one; a point outside has two tangents.