Tangent to a Circle

Introduction to Tangent

Definition

A tangent to a circle is a straight line that touches the circle at exactly one point, known as the point of contact. The tangent is unique in that it does not cross or pass through the circle, unlike a secant, which intersects the circle at two points.

Key Properties of Tangents

Perpendicularity:

1. - The tangent is always perpendicular to the radius drawn to the point of contact.
  - Mathematically, if $\boldsymbol$ is the center of the circle, $\boldsymbol$ is the point of contact, and $\boldsymbol$ is the tangent, then: $\boldsymbol$

Tangency:

1. - The tangent touches the circle at only one point and does not pass through the interior of the circle.

A Tangent Touches the Circle at Only One Point:

1. - A tangent line intersects the circle at a single, unique point known as the point of contact.
  - Unlike a secant, which intersects the circle at two points, the tangent remains external to the circle while touching it at one point.

A Tangent is Always Perpendicular to the Radius at the Point of Contact:

1. - For a circle with center $\boldsymbol$ , if $\boldsymbol$ is the point of contact and $\boldsymbol$ is the tangent, then: ∠OPB=90∘.\boldsymbol.∠OPB=90∘.
  - This property is fundamental and forms the basis for solving many geometric problems involving tangents.

From an External Point, Exactly Two Tangents Can be Drawn to a Circle:

1. - When a point lies outside a circle, it is possible to draw exactly two tangents to the circle from that point.
  - The points of tangency on the circle divide it into two parts, with each tangent forming a unique line to the external point.

The Lengths of Tangents Drawn from an External Point to a Circle are Equal:

- If $\boldsymbol$ is an external point, and $\boldsymbol$ and $\boldsymbol$ are tangents drawn to a circle with center $\boldsymbol$ and points of tangency $\boldsymbol$ and $\boldsymbol$ , then: PA=PB.\boldsymbol.PA=PB.
- This property is proven using triangle congruence (CPCT) and is widely used in solving problems related to tangents.

Relationship Between Tangents and Secants

Definition of a Secant

A secant is a line that intersects a circle at two distinct points.
For example, a line $\boldsymbol$ intersects a circle at points $\boldsymbol$ and $\boldsymbol$ , making it a secant.

Definition of a Tangent

A tangent is a line that intersects the circle at exactly one point, called the point of contact.

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 of contact.
- At this stage, the secant transforms into a tangent.

Key Observations

Both tangents and secants are external to the circle, but their interaction differs:
- A secant always cuts through the circle at two points.
- A tangent just touches the circle at one point without cutting through.
The perpendicularity property of tangents does not apply to secants.

Theorem 1: Tangent is Perpendicular to the Radius

Statement

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

If $\boldsymbol$ is the center of the circle, $\boldsymbol$ is the point of tangency, and $\boldsymbol$ is the tangent at $\boldsymbol$ , then:

$\boldsymbol$

Proof

Given:

A circle with center $\boldsymbol$ .
A tangent $\boldsymbol$ touches the circle at point $\boldsymbol$ .

To Prove:

$\boldsymbol$

Construction:

Assume $\boldsymbol$ is the center of the circle, and $\boldsymbol$ is the radius.
Suppose $\boldsymbol$ is not perpendicular to $\boldsymbol$ . Construct a line $\boldsymbol$ perpendicular to $\boldsymbol$ such that $\boldsymbol$ lies on $\boldsymbol$ .

Proof Steps:

Assumption of Shortest Distance:
- The radius $\boldsymbol$ is the shortest distance from the center $\boldsymbol$ to the line $\boldsymbol$ .
- By the property of perpendicular lines, $\boldsymbol$ should be the shortest distance if $\boldsymbol$ .
Contradiction:
- If $\boldsymbol$ is shorter than $\boldsymbol$ , then $\boldsymbol$ cannot be on the circle because the radius must be constant.
- This contradicts the definition of a circle.
Conclusion:
- Hence, $\boldsymbol$ , proving that the tangent is perpendicular to the radius.

Theorem 2: Lengths of Tangents

Statement

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

If $\boldsymbol$ is an external point and $\boldsymbol$ and $\boldsymbol$ are tangents drawn to a circle with center $\boldsymbol$ , then:

$\boldsymbol$

Proof

Given:

A circle with center $\boldsymbol$ and radius $\boldsymbol$ .
An external point $\boldsymbol$ outside the circle.
Two tangents $\boldsymbol$ and $\boldsymbol$ drawn to the circle, touching it at points $\boldsymbol$ and $\boldsymbol$ .

To Prove:

$\boldsymbol$

Proof Steps:

Perpendicularity of Tangent and Radius:
- By Theorem 10.1, $\boldsymbol$ and $\boldsymbol$ .
Triangles Involved:
- Consider the triangles $\boldsymbol$ and $\boldsymbol$ .
Congruence of Triangles:
- In $\boldsymbol$ and $\boldsymbol$ :
  - $\boldsymbol$ (radii of the same circle).
  - $\boldsymbol$ (common side).
  - $\boldsymbol$ (tangent is perpendicular to the radius).
- By RHS congruence criterion, $\boldsymbol$ .
Corresponding Parts of Congruent Triangles (CPCT):
- From congruence, $\boldsymbol$ .

Conclusion

The tangents $\boldsymbol$ and $\boldsymbol$ from the external point $\boldsymbol$ to the circle are equal in length.

Applications

Calculating Tangent Lengths:
- Given the radius and the distance from the center to the external point, tangent lengths can be determined using the Pythagoras theorem.
- Example: If $\boldsymbol$ and $\boldsymbol$ , then: $\boldsymbol$
Geometric Problems:
- Used to solve problems involving circles, tangents, and distances in coordinate geometry and construction.