Circles: Tangents and Secants

Secant: Definition and Properties

In geometry, a secant is a fundamental concept related to circles. It is defined based on how a line interacts with a circle on a plane.

A secant to a circle is formally defined as a straight line that intersects the circle at exactly two distinct points. The term "secant" comes from the Latin word 'secare', which means 'to cut', aptly describing its function of cutting through the circle.

As shown in the figure above, the line L passes through the interior of the circle and meets its circumference at two separate points, labelled here as A and B. Because line L intersects the circle at these two distinct points, it is classified as a secant line.

Properties and Distinctions

Understanding the definition of a secant involves appreciating its characteristics and distinguishing it from other types of lines related to a circle, such as chords and tangents.

Firstly, a secant is an infinite line, meaning it extends without end in both directions. This is a crucial distinction from a chord. A chord, on the other hand, is the finite line segment connecting the two points where a secant intersects the circle (like the segment AB in the figure). The chord is contained within the secant line and lies entirely inside or on the circle.

It follows that every secant line contains a chord of the circle – specifically, the chord defined by its two points of intersection with the circle.

A special case arises with diameters. A diameter is the longest chord of a circle, passing through the centre. When a diameter is extended indefinitely in both directions to form a line, this line is a type of secant that passes through the circle's centre. Thus, every line containing a diameter is a secant.

Unlike a tangent, which touches the circle at only one point, a secant cuts through the interior of the circle, entering at one point of intersection and exiting at the other. This passage through the interior is characteristic of a secant.

Answer:

Tangent: Definition and Relation with Radius at Point of Contact (Theorem)

Apart from secants, lines can interact with a circle in another unique way – by just 'touching' it at a single point. Such a line is called a tangent.

Definition

A tangent to a circle is defined as a straight line in the plane of the circle that intersects the circle at exactly one point. It is the limiting case of a secant where the two points of intersection coincide.

The specific point where the tangent line meets the circle is of particular significance and is referred to as the point of contact or the point of tangency.

In the illustration above, the line M touches the circle at point P and at no other point. Therefore, line M is the tangent to the circle at P, and P is the point of contact.

The word 'tangent' has its roots in the Latin word 'tangere', which means 'to touch'. This etymology perfectly captures the nature of a tangent line in relation to a curve.

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

This is a fundamental theorem in circle geometry that establishes a crucial relationship between a tangent line and the radius drawn to its point of tangency.

Given: A circle with centre O and radius $r$. A tangent line L touches the circle at point P. OP is the radius of the circle connecting the centre O to the point of contact P.

To Prove: The radius OP is perpendicular to the tangent line L at the point of contact P. Symbolically, $OP \perp L$.

Proof: We can prove this theorem using the property that the shortest distance from a point to a line is the perpendicular distance.

Let P be the point where the tangent line L touches the circle with centre O.

Consider any point Q on the tangent line L, other than the point of contact P. So, $Q \neq P$.

By the definition of a tangent, the line L intersects the circle at P only. This means that any other point Q on the line L cannot be on the circle. If Q were on the circle (and Q $\neq$ P), then the line L would intersect the circle at two points (P and Q), making it a secant, which contradicts our assumption that L is a tangent.

Therefore, any point Q on the tangent line L, other than P, must lie outside the circle.

Since Q lies outside the circle, its distance from the centre O must be greater than the radius of the circle, which is OP.

This inequality, OQ $>$ OP, holds true for every single point Q on the tangent line L, with the sole exception of the point P itself, for which the distance from O is exactly the radius ($OP = r$).

This observation implies that the line segment OP represents the shortest distance from the centre O to any point on the tangent line L.

A fundamental geometric principle states that the shortest distance from an external point to a line is always measured along the perpendicular from the point to the line.

Since OP is the shortest distance from the point O to the line L, it must be perpendicular to the line L.

Thus, the radius drawn to the point of contact of a tangent is perpendicular to the tangent line at that point.

Hence, the theorem is proved.

Corollaries derived from the theorem:

• Uniqueness of Tangent: At any given point on a circle, there exists one and only one tangent line. If there were two distinct tangent lines at the same point, both would have to be perpendicular to the same radius at the same point, which is geometrically impossible.
• Normal to the Circle: The line which is perpendicular to the tangent at the point of contact is called the normal to the circle at that point. From the theorem, the radius through the point of contact lies along the normal. Thus, the line containing the radius through the point of contact is the normal to the circle at that point.

Answer:

Given:

• Centre of the circle is O.
• Tangent is AB, touching the circle at P.
• Radius OP = 8 cm.
• Length of the tangent segment from A to P is AP = 15 cm.

We need to find the distance OA.

According to the theorem "The tangent at any point of a circle is perpendicular to the radius through the point of contact," the radius OP is perpendicular to the tangent line AB at the point of contact P.

This means the angle $\angle \text{OPA}$ is a right angle ($90^\circ$).

$\angle \text{OPA} = 90^\circ$

(Tangent is perpendicular to radius at point of contact)

Now consider the triangle $\triangle \text{OPA}$. Since $\angle \text{OPA} = 90^\circ$, $\triangle \text{OPA}$ is a right-angled triangle with the hypotenuse OA.

We can use the Pythagorean theorem in $\triangle \text{OPA}$:

$OA^2 = OP^2 + AP^2$

(Pythagorean theorem)

Substitute the given values:

$\text{OA}^2 = (8 \text{ cm})^2 + (15 \text{ cm})^2$

$\text{OA}^2 = 64 \text{ cm}^2 + 225 \text{ cm}^2$

$\text{OA}^2 = 289 \text{ cm}^2$

To find OA, take the square root of both sides:

$\text{OA} = \sqrt{289} \text{ cm}$

We know that $17^2 = 289$.

$\text{OA} = 17 \text{ cm}$

The distance of point A from the centre O is 17 cm.

Number of Tangents from a Point on/outside/inside a Circle

The position of a given point relative to a circle significantly determines how many tangent lines can be drawn from that point to the circle. There are three distinct cases based on whether the point is inside, on, or outside the circle.

Case 1: Point Inside the Circle

If a point P is located in the interior of a circle, it is not possible to draw any tangent line to the circle that passes through P. The number of tangents from a point inside the circle is zero.

Reason: By definition, a tangent intersects a circle at exactly one point. If a line passes through a point that is strictly inside the circle, it must inevitably cross the circle's boundary twice to exit the circle (forming a chord and thus being part of a secant line). No line passing through an interior point can 'just touch' the circle at a single point.

Case 2: Point On the Circle

If a point P lies exactly on the circumference of the circle, there is precisely one tangent line that can be drawn to the circle at point P.

Reason: As established by the theorem in the previous section, the tangent at any point P on the circle is the unique line perpendicular to the radius OP at P. There can be only one line passing through P that is perpendicular to the radius OP. Therefore, only one tangent exists at any given point on the circle.

Case 3: Point Outside the Circle

If a point P is situated in the exterior of the circle, it is possible to draw precisely two tangent lines from P to the circle.

Reason: From any external point P, two distinct lines can be drawn such that each line touches the circle at exactly one point. Let these points of contact be A and B. The line segment PA is a tangent segment from P to the circle at A, and the line segment PB is a tangent segment from P to the circle at B. The lines containing these segments, extending indefinitely, are the two tangent lines from P to the circle. These are the only two lines passing through P that satisfy the definition of a tangent.

The points A and B are called the points of contact of the tangents from P.

Length of Tangents Drawn from an External Point (Theorem)

When we consider an external point from which two tangents are drawn to a circle, a significant property emerges concerning the lengths of these tangent segments. The length of the tangent from an external point P to a circle is defined as the distance from P to the point where the tangent touches the circle.

In the diagram shown, P is a point located outside the circle with centre O. Two tangents are drawn from P, touching the circle at points A and B respectively. The line segments PA and PB are the tangent segments from P. Their lengths, denoted by $PA$ and $PB$, are referred to as the lengths of the tangents from point P to the circle.

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

This theorem states that no matter where the external point P is, as long as it is outside the circle, the distance from P to the two points of contact will be exactly the same ($PA = PB$).

Given: A circle with centre O. P is an external point. PA and PB are the two tangents drawn from P to the circle, touching the circle at points A and B respectively.

To Prove: $PA = PB$.

Construction: To aid the proof, we connect the centre O to the external point P, and also connect the centre O to the points of contact A and B. This forms the line segments OA, OB, and OP.

Proof:

We know from the theorem "The tangent at any point of a circle is perpendicular to the radius through the point of contact" that the radius drawn to the point of contact is perpendicular to the tangent at that point.

Therefore, in our diagram:

Now, consider the two triangles $\triangle \text{OAP}$ and $\triangle \text{OBP}$. Both are right-angled triangles (at A and B respectively).

We can compare these two triangles based on their sides and angles:

Since we have a right angle, the hypotenuse, and one side of one triangle equal to the corresponding parts of the other triangle, we can conclude that the two triangles are congruent by the RHS (Right angle-Hypotenuse-Side) congruence criterion.

$\triangle \text{OAP} \cong \triangle \text{OBP}$ (By RHS congruence rule)

Since the triangles are congruent, their corresponding parts are equal (CPCT - Corresponding Parts of Congruent Triangles).

Therefore, the corresponding sides PA and PB must be equal.

This proves that the lengths of the tangents drawn from an external point to a circle are equal.

Other Properties from Congruence

The congruence of triangles $\triangle \text{OAP}$ and $\triangle \text{OBP}$ also provides us with other important properties relating to the angles formed by the tangents and the line joining the external point to the centre:

1. Tangents are equally inclined to the line segment joining the point to the centre:

   $\angle \text{APO} = \angle \text{BPO}$

   (CPCT)

   This implies that the line segment OP, which connects the centre O to the external point P, bisects the angle formed by the two tangent segments ($\angle \text{APB}$).

2. Line joining the centre to the external point bisects the angle between the radii to the points of contact:

   $\angle \text{AOP} = \angle \text{BOP}$

   (CPCT)

   This implies that the line segment OP also bisects the angle subtended by the chord of contact AB at the centre of the circle ($\angle \text{AOB}$). The chord AB connecting the points of contact A and B is called the 'chord of contact'.

3. OP is the perpendicular bisector of the chord of contact AB:

   $\angle \text{AOM} = \angle \text{BOM}$

   (CPCT, where M is intersection of OP and AB)

   AM $=$ BM

   (CPCT)

   $\angle \text{AMO} = \angle \text{BMO} = 90^\circ$

   (CPCT, implies perpendicularity)

   This means that the line segment OP is the perpendicular bisector of the chord of contact AB.

Answer:

Given:

• A circle with centre O.
• P is an external point.
• PT and PS are tangents drawn from P to the circle, touching at T and S respectively.
• Length of tangent PT = 8 cm.

We need to find the length of PS.

According to the theorem "The lengths of tangents drawn from an external point to a circle are equal," the lengths of the two tangent segments from an external point to a circle are equal.

Therefore,

PS $=$ PT

(Lengths of tangents from external point P)

Since PT = 8 cm,

PS $=$ 8 cm

The length of the tangent PS is 8 cm.

Example 2. In the figure, if TP and TQ are two tangents to a circle with centre O such that $\angle \text{POQ} = 110^\circ$, then find $\angle \text{PTQ}$.

Answer:

Properties of Parallel Tangents

When considering multiple tangents to a circle, an interesting geometric situation arises when two such tangents are parallel to each other. This configuration has a significant property related to the points where these parallel tangents touch the circle.

In the figure shown, line L is a tangent to the circle at point A, and line M is a tangent to the circle at point B. If these two tangent lines L and M are parallel to each other ($L \parallel M$), then the points of contact A and B are related in a special way.

Theorem: The points of contact of two parallel tangents to a circle lie on a diameter.

This theorem can also be stated as: The line segment connecting the points of contact of two parallel tangents to a circle is a diameter of the circle.

Given: A circle with centre O. Two parallel tangent lines, L and M, touch the circle at points A and B respectively ($L \parallel M$).

To Prove: The line segment AB passes through the centre O, meaning AB is a diameter of the circle.

Construction: Join the centre O to the points of contact A and B, forming the radii OA and OB. (We do not need any additional lines for this proof using the standard approach).

Proof:

We know that the radius drawn to the point of contact is perpendicular to the tangent at that point.

Since L is the tangent at point A and OA is the radius through A, we have:

Similarly, since M is the tangent at point B and OB is the radius through B, we have:

We are given that the tangent lines L and M are parallel ($L \parallel M$).

Now consider the line containing the radius OA. This line is perpendicular to L. Since L is parallel to M, any line perpendicular to L must also be perpendicular to M.

So, the line containing OA is perpendicular to M. But we also know that the line containing OB is perpendicular to M (at point B).

There is only one line perpendicular to M at point B. This unique line must pass through O (since OB is on this line). This means that the line containing OA must also pass through B and be perpendicular to M at B.

For the line containing OA to pass through B, the points O, A, and B must be collinear. Specifically, since A and B are on the circle and O is the centre, the line segment AB must pass through the centre O.

Therefore, AB is a diameter of the circle.

Hence, the points of contact of two parallel tangents to a circle lie on a diameter.

Converse Theorem: Tangents drawn at the ends of a diameter of a circle are parallel.

This is the reverse statement: if we draw tangents at the two endpoints of a diameter, these two tangent lines will be parallel to each other.

Given: A circle with centre O. AB is a diameter of the circle. Tangent line L is drawn at endpoint A, and tangent line M is drawn at endpoint B.

To Prove: Tangent L is parallel to Tangent M ($L \parallel M$).

Proof:

Since L is the tangent at point A, and OA is the radius through A, we know that the tangent is perpendicular to the radius at the point of contact.

This implies that the angle between the radius OA and the tangent L is $90^\circ$. Let X be a point on tangent L such that X and B are on opposite sides of the diameter AB. Then the angle $\angle \text{OAX} = 90^\circ$. Since O lies on the line segment AB (because AB is a diameter), the angle $\angle \text{BAX} = 90^\circ$.

Similarly, since M is the tangent at point B and OB is the radius through B, we have:

This implies that the angle between the radius OB and the tangent M is $90^\circ$. Let Y be a point on tangent M such that Y and A are on opposite sides of the diameter AB. Then the angle $\angle \text{OBY} = 90^\circ$. Since O lies on the line segment AB, and OB is along the line BA (part of the diameter), the angle $\angle \text{ABY} = 90^\circ$.

Now consider the lines L and M intersected by the transversal line AB (which is the diameter).

From (i) and (ii), we have $\angle \text{BAX} = 90^\circ$ and $\angle \text{ABY} = 90^\circ$. These angles, $\angle \text{BAX}$ and $\angle \text{ABY}$, are alternate interior angles formed by the transversal AB intersecting lines L and M.

When a transversal intersects two lines such that the alternate interior angles are equal, the two lines are parallel.

Therefore, the tangent line L is parallel to the tangent line M ($L \parallel M$).

Hence, tangents drawn at the ends of a diameter of a circle are parallel.

Answer:

Let the circle have centre O. Let the two parallel tangents be L and M, touching the circle at points A and B respectively.

According to the theorem "The points of contact of two parallel tangents to a circle lie on a diameter," the line segment joining the points of contact A and B must be a diameter of the circle.

The distance between two parallel lines is the perpendicular distance between them. The diameter AB is perpendicular to both parallel tangents L and M (since the radius to the point of contact is perpendicular to the tangent, and AB passes through the centre O, it is perpendicular to both L at A and M at B).

Therefore, the distance between the parallel tangents L and M is equal to the length of the diameter AB.

Given that the distance between the parallel tangents is 10 cm.

Diameter AB $=$ Distance between parallel tangents

Diameter $=$ 10 cm

The radius of a circle is half of its diameter.

Radius $(r)$ $=$ $\frac{1}{2} \times$ Diameter

$r = \frac{1}{2} \times 10 \text{ cm}$

$r = 5 \text{ cm}$

The radius of the circle is 5 cm.