Tangent to a Circle: Equations, Theorems & Examples

A tangent to a circle, in Euclidean plane geometry, refers to a line that touches the circle at exactly one point and never enters the circle’s interior. It plays an important role in many geometrical constructions as well as proofs and forms the subject of many theorems. 

• Tangent to a circle is perpendicular to the radius of the circle. 
• It takes into consideration radial lines and orthogonal circles.
• These lines cannot be drawn through a point within a circle.
• The geometrical shape of circles and tangent lines form a reflection of symmetry.

Many geometrical transformations, such as rotations, scalings, translations, inversions, and map projections, exhibit this property. 

Key Terms: Tangent to a Circle, Tangent line, Circle, Plane Geometry, Geometry, Geometrical constructions, Tangency, Curved line, Reflection of Symmetry, Radius, Tangent Properties, Tangent Theorems

---

Tangent to a Circle Definition

[Click Here for Sample Questions]

A tangent to a circle is a line that intersects the circle at a single point. The point of tangency is where the tangent meets the circle. The tangent is perpendicular to the circle's radius, which it crosses.

• The word tangent is derived from the Latin word tangere, which means "to touch."
• A tangent line (or tangent) is a line or plane that meets a curved line or surface at a single point in geometry.
• The line that contains the radius through the point of contact is referred to as being 'normal' to the circle at the point.
• Only one point on the tangent intersects the circle.

• Consider the above figure, which shows a circle with O in its centre. 
• The tangent of a circle passing through point R is PQ.
• Join OS at a point S on a tangent PQ other than R. 
• Point S should be outside the circle because if it is inside the circle, PQ will be a secant to the circle and not a tangent.

As a result, the OS will be greater than the circle's radius OR. Except for the point of contact R, this occurs at every point on PQ.

• The shortest distance between the centre of circle O and the tangent PQ is determined to be OR.
• OR is perpendicular to PQ because the shortest distance between a point and a line is the perpendicular distance between them.

Also Read:

---

Condition of Tangency

[Click Here for Sample Questions]

When a tangent contacts a curve at a single point, it is termed a tangent to a circle; otherwise, it is merely a line. We can define the requirements for tangent as follows, based on the point of tangency and its location in relation to the circle:

When the point lies outside the circle

We can conclude from the above illustration that there are exactly two tangents to the circle from a point outside the circle.

Outside the Circle

When the point lies inside the circle

Consider the point P inside the circle in the illustration above; all of the lines passing through P intersect the circle twice. It follows that no tangent to a circle can be formed that passes through a point inside the circle.

Inside the Circle

When the point lies on the circle

According to the illustration, there is only one tangent to a circle that passes through a point on the circle.

Lies on the Circle

---

Tangent Properties

[Click Here for Sample Questions]

• The tangent to a circle always makes a single point of contact with the circle.
• At the point of tangency, it is perpendicular to the circle's radius.
• It never has two points where it intersects the circle.
• Tangents from an exterior point to a circle have the same length.

---

General Equations

[Click Here for Sample Questions]

The general equations of tangent to a circle are as follows:

• At (x₁, y₁), the tangent to the circle equation x²+ y²=a² is

xx₁+yy₁= a².

• At (x₁, y₁), the tangent to the circle equation x²+ y²+2gx+2fy+c =0 is

xx₁+yy₁+g(x+x₁)+f(y +y₁)+c =0 

• At (a cos θ, a sin θ), the tangent to the circle equation x²+ y²=a² is 

x cos θ+y sin θ= a 

• For a line y = mx +c, the tangent to a circle equation x²+ y²=a² is

y = mx ± a√ [1+ m²].

---

Tangent Formula

[Click Here for Sample Questions]

Assume P is located outside the circle. We can draw two tangents to the circle from point P, which intersects at points A and B. Now draw a secant from P to Q and R, intersecting the circle. 

• The tangent line from point P to point S is denoted by PS. 
• It is also known as Tangent Secant theorem.
• The formula for the tangent and secant of the circle is now:

PR/PS = PS/PQ.

PS²=PQ.PR

---

Tangent Theorems

[Click Here for Sample Questions]

The theorem of tangents to a circle are as follows:

Theorem 1

The theorem 1 is as follows:

Statement: At the point of contact, the tangent to the circle is perpendicular to the radius of the circle.

Given: It is given that tangent AB to a given circle S (with the centre O), and the point of contact is C.

Prove: It is required to be proved that OC is perpendicular to the tangent AB

Proof: As seen in the figure, point D is seen lying outside the circle. When we join OD, we will get OD > OC, which is the radius of a circle. This condition is applicable to every point on line AB except point C.

• Thus, proving OD > OC, it is shown that OC is the shortest of all points O to the other points on AB.
• Hence proved

Theorem 1

Theorem 2

The theorem 2 is as follows:

Statement: When two tangents are drawn from the circle's external point, they are of equal length.

Proof:  The theorem will be proved if we can show that ΔCAO is congruent to ΔCBO. When we compare these two triangles, it can be concluded that 

• OA = OB (radii of a similar circle)
• OC = OC (common side of the figure)
• ∠OAC = ∠OBC = 90° (Tangent drawn to a circle is perpendicular to the radius at the point of tangency)

Thus proving that ΔCAO is congruent to ΔCBO.

Theorem 2

Also Read:

---

Things To Remember 

• Tangent to a circle represents the line segments that touch the given circle only at one particular point. 
• It is determined that circles can have many tangents.
• Thales theorem, also known as basic proportionality theorem, is used for the construction of tangent lines.
• The most common example of the tangent to a circle includes the velocity of a particle moving in a uniform circular motion.
• The point at which the line touches the curve of the circle is called the point of tangency.