Introduction To Circles

Introduction to Circles

Definition of a Circle

A circle is a two-dimensional geometric shape that consists of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius.

Center ( $\boldsymbol{O}$ ):
The fixed point from which the circle is defined.
Radius ( $\boldsymbol{r}$ ):
The constant distance from the center to any point on the circle.

Interaction Between a Line and a Circle

When a straight line is drawn in relation to a circle, there are three possible interactions:

Non-Intersecting Line (External Line):
- The line does not touch or pass through the circle at any point.
- Example: In the diagram, line $\boldsymbol{PQ}$ lies completely outside the circle and does not have any common points with it.
- Key Observation:
  Such a line is at a distance greater than the radius from the center of the circle.
Secant:
- A secant is a line that intersects the circle at two distinct points.
- Example: In the diagram, line $\boldsymbol{PQ}$ passes through the circle, intersecting it at points $\boldsymbol{A}$ and $\boldsymbol{B}$ .
- Key Properties of a Secant:
  - It divides the circle into two parts.
  - The chord $\boldsymbol{AB}$ is the line segment formed between the two points of intersection.
Tangent:
- A tangent is a line that touches the circle at exactly one point.
- Example: In the diagram, line $\boldsymbol{PQ}$ touches the circle at point $\boldsymbol{T}$ .
- Key Properties of a Tangent:
  - The tangent is perpendicular to the radius at the point of contact ( $\boldsymbol{\angle OTP = 90^\circ}$ ).
  - A tangent never passes through the circle.
  - There can be only one tangent at a given point on the circle.

Real-Life Analogy

Non-Intersecting Line: A rope lying outside a circular well, not touching the well.
Secant: A knife cutting through a circular cake, forming two intersection points.
Tangent: The edge of a wheel touching the road at one point as it rolls.

Tangent to a Circle

Definition

A tangent to a circle is a straight line that touches the circle at exactly one point without passing through it. This unique point where the tangent touches the circle is called the point of contact.

Key Property: The tangent is always perpendicular to the radius drawn to the point of contact.
For a circle with center $\boldsymbol{O}$ and a tangent $\boldsymbol{AB}$ touching the circle at point $\boldsymbol{P}$ , we have: $\boldsymbol{\angle OPB = 90^\circ}$

Understanding Tangents

Uniqueness of Tangents at a Given Point:
- At any given point on a circle, only one tangent can be drawn.
- No two distinct tangents can pass through the same point of contact on the circle.
Relationship Between Tangents and Secants:
- A tangent can be thought of as a limiting case of a secant:
  - As the two points of intersection of a secant move closer together, they eventually merge into a single point, forming a tangent.
- Secants intersect the circle at two points, while tangents intersect it at exactly one point.

Examples from Real Life

Pulley System:
- In a pulley used to draw water from a well, the rope is tangential to the circular edge of the pulley at the point where the force is applied.
Bicycles and Cars:
- The wheel of a bicycle or car touches the road at one single point at any given time. This point is the tangent between the wheel and the surface.
Rolling Circular Objects:
- The contact between any rolling circular object and the ground is an example of a tangent, as the circular edge touches the ground at a single point.

Properties of Tangents

Theorem 1:

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

Theorem 2: Lengths of Tangents

The lengths of tangents drawn from an external point to a circle are equal.

Activities to Explore Tangents

Activity 1:
- Rotate a straight wire around a circular object. Observe how the wire touches the circle at only one point, forming a tangent.
- Helps understand the uniqueness of the tangent at any given point.
Activity 2:
- Draw lines parallel to a secant, reducing their distance from the circle. As the lines approach tangency, the points of intersection reduce to one.
- Demonstrates the transition from a secant to a tangent.
Activity 3:
- Explore tangents by drawing them from:
  - Inside the circle: No tangents possible.
  - On the circle: Only one tangent possible.
  - Outside the circle: Exactly two tangents can be drawn.
- Highlights the different cases and the uniqueness of tangents.

Introduction To Circles