What is a tangent to a circle?

It's a line that just touches the circle at one point and is perpendicular to the radius at that point.

How many tangents can be drawn from an external point to a circle?

Two tangents can be drawn from an external point to a circle.

Can the tangent cut the circle at two points?

No, the tangent touches the circle at one point.

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Tangent in Mathematics

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Tangent in Mathematics

Master Circle Tangents & Theorems in Minutes: Learn how a straight line interacts with a circle at exactly one point. Master the fundamental properties of the point of contact, understand the two major board exam theorems, and solve high-yield geometric problems easily—all formatted in clean, plain text for simple copy-pasting.

Class: 10 Mathematics (CBSE)
Chapter: Circles
Estimated Learning Time: 15–20 Minutes

1.0Learning Outcomes

After completing this lesson, you will be able to:

Distinguish between a Tangent, a Secant, and a Chord of a circle.
State and apply Theorem 1 (Radius is perpendicular to the tangent at the point of contact).
State and apply Theorem 2 (Tangents drawn from an external point are equal in length).
Use the Pythagorean theorem to solve numerical problems involving tangent lengths.

Tangent in Mathematics

Tangents are the most common concept of two vital topics of mathematics, geometry and trigonometry, having a different meaning and use in each one. Whether it's the sophisticated flair of a line to a curve in geometry, or how angles and ratios are brought to life in trigonometry, tangents are everywhere. So, let's explore how this simple yet profoundly insightful idea is central to everything, from circle theorems to the very structure of trigonometric identities.

2.0Tangents in Geometry

In geometry, the tangent is primarily interested in lines that are tangent to curves, such as circles, and in determining relationships between points, angles, and lengths. Tangents in geometry are used in various aspects of geometry with different meanings in different contexts, such as:

Tangent to a Circle

A tangent to a circle is a line that is straight and touches the circle at a single point. The point where the tangent touches the circle is referred to as the point of tangency or the point of contact. Tangents are unique to a certain point, meaning no two tangents are drawn from a single point of contact.

Theorems on Tangents for Circles

Tangents in geometry follow several theorems, especially when dealing with circles. These theorems form the basis of many proofs and constructions in other topics of geometry:

The Tangent-Radius Theorem:

A tangent to a circle is always at right angles to the radius drawn to the point of contact.

If a line is touching a circle at point P and O is the centre, then OA is tangent.

Two Tangents from an External Point:

From any external point of a circle, two tangents can be drawn exactly, and both are of equal length.

If PA and PB are tangents from P to a circle with points of contact A and B, then PA = PB.

Image showing Two Tangents from an External Point

Tangent to a Curve

A tangent to a curve is a straight line that touches the curve at one and only one point and does not cut across it. The tangent line best approximates the curve linearly at the point. The most important property of the tangent is that it is the instantaneous rate of change (slope) of the curve at the point of tangency.

3.0Tangent Formula and Derivation of Tangent Formula

The tangent formula is derived from the tangent secant theorem, which states, if given a circle with centre O, PQ be a tangent to this circle with external point P, and PRS be the secant to the same circle with R and S being the points on the circle. Then, according to this theorem, the tangent formula can be written as:

$P Q^{2} = PS \times PR$

Derivation of Tangent Formula

As per the figure given below, where O is the centre of the circle, PRS is a secant of the circle, and PQ is the tangent to the circle. A line OA is drawn perpendicular to RS. Now, Join OR, OP and OQ.

From the construction we know, OP ⟂ PQ, hence,

RA = AS …… (1) (A perpendicular drawn from the centre of the circle on the chord bisects the chord)

PR = (PA – RA) and PS = (PA + AS)

Now, Multiply PR with PS

PR × PS = (PA – RA) (PA + AS)

⇒ PR × PS = (PA – RA) (PA + RA) [From equation 1]

⇒ PR × PS = PA² – RA² ….(2)

In △ POA, by Pythagoras' theorem

OP² = OA² + PA²

⇒ PA² = OP² – OA²

From equation 2

⇒ PR × PS = OP² – OA² – RA²

⇒ PR × PS = OP² – (OA² + RA²) …..(3)

Now, similarly in △ ROA

OR² = OA² + RA²

⇒ RA² = OR² – OA²

From equation 3

⇒ PR × PS = OP² – (OA² + OR² – OA²)

⇒ PR × PS = OP² – OA² – OR² + OA²)

⇒ PR × PS = OP² – OR² …. (4)

Since OR = OQ (Radii of the circle)

Thus, equation 4 becomes

PR × PS = OP² – OQ² ….. (5)

In △ POQ

OP² = OQ² + PQ²

⇒ PQ² = OP² – OQ²

From equation 5

PQ² = PR × PS

4.0Trigonometric Interpretation of Tangents

Now that we know about the tangents in geometry, let’s take a quick look at the tangents of trigonometry. In trigonometry, the term tangent is related to the right-angled triangle. Tangents are the functions that play a critical role in finding the angular measurement of these triangles, and are known as the trigonometric ratios. The formula to find these tangents is:

$t an θ = \frac{Perpendicular}{Base}$

Here is the angle between the perpendicular and the base of the right-angled triangle. The numerical values for this tan are different for different angles. For example:

Tangent 0°	0
Tangent 30°	$\frac{1}{3}$
Tangent 45°	1
Tangent 60°	$3$
Tangent 90°	∞

5.0Equation of Tangent

The equation of a tangent to a curve at a given point can be determined using the concept of slope. It is represented by the derivative of the function. The general idea behind this is that the equation of the straight line with slope m passing through point (x₀, y₀) is:

$y - y_{0} = m (x - x_{0})$

For a curve defined by y = f(x), the slope of the tangent at a specific point x₀ is given by the derivative:

$m = f^{'} (x_{0})$

Hence, the equation of the tangent line to the curve y = f(x) can be written as:

$y - y_{0} = f^{'} (x_{0}) (x - x_{0})$

In a trigonometric context, this slope “m” can be related to an angle θ, which is the angle of inclination of the tangent line made with the positive x-axis. The relationship is given by:

$t an θ = f^{'} (x_{0})$

Hence, we conclude that the slope of a tangent line can be written as:

Slope of the tangent = $t an θ$

6.0Application of Tangents

Tangents have significant applications in different fields of study, which include:

Geometry: Building shapes and establishing theorems.
Engineering: Creating roads, bridges, and railroads.
Physics: Determining angles, forces, and motion rates.
Computer Graphics: Smoothing curves and edges.
Architecture: Building designs involving curves and circular structures.

7.0Solved Examples in Tangents

Problem 1: From a point P outside a circle, two tangents PA and PB are drawn to the circle. If PA = 7 cm, what is the length of PB?

Solution: Given that PA and PB are two tangents from an external point P.

We know that from any external point of a circle, two tangents can be drawn exactly, and both are of equal length. Hence,

PA = PB = 7cm

Problem 2: Find the equation of the tangent to the curve y = x² at the point where x = 2.

Solution: given that y = x²

Put x = 2, y = 2² = 4.

Point (2, 4)

Now, y’(x) = 2x

y’(x) = 4

Using the equation of the tangent, we have;

$y - y_{0} = y^{'} (x_{0}) (x - x_{0})$

y-4=4(x-2)

y-4=4x-8

4x-y=4

Problem 3: If a tangent to a curve at a point makes an angle of 30 degrees with the x-axis, what is the slope of the tangent?

Solution: Given that the angle of tangent to a curve is 30°, we know:

Slope of the tangent= $t an θ$

Hence, Slope of the tangent= $t an 30 = \frac{1}{3}$

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9.0Supporting Study Materials

This study material CBSE Notes and NCERT Solutions for the Chapter "Circles" on Tangents in Circles Topics is designed according to the latest CBSE Class 10 Mathematics syllabus and NCERT guidelines. It provides clear explanations of key concepts, definitions, theorems, and important questions to help students understand the geometric properties of a tangent, theorems regarding tangents drawn from an external point, the point of contact, and prepare effectively for examinations.

CBSE Class 10 Maths Notes Chapter 10 Circles	NCERT Solutions for Class 10 Maths Chapter 10: Circles

10.030-Second Revision

Tangent = A straight line touching a circle at exactly one point.
Point of Contact = The single shared spot where the line kisses the circle.
Radius-Tangent Rule (Theorem 1) = Radius meets the tangent at a perfect 90-degree angle.
External Tangents Rule (Theorem 2) = Two tangents from the same outside point are always equal in length (PA = PB).

The Right-Triangle Formula = Connect the center (O), the contact point (A), and the external point (P) to use: $O P^{2} = O A^{2} + P A^{2}$
11.0PREVIOUS YEAR QUESTIONS (PYQs)
Q1. From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the center of the circle is 25 cm. Find the radius of the circle. (CBSE Board)
Solution
Step 1: Visualize the geometry and identify components.
Let O be the center of the circle, and P be the point of contact of the tangent drawn from external point Q.
Length of the tangent (PQ) = 24 cm
Distance from center to external point (OQ) = 25 cm
Radius of the circle (OP) = ?
Step 2: Apply Theorem 1.
According to Theorem 1, the radius OP is perpendicular to the tangent PQ at the point of contact P. Therefore, triangle OPQ is a right-angled triangle where the hypotenuse is OQ.
Step 3: Solve using the Pythagorean theorem.

$O Q^{2} = O P^{2} + P Q^{2}$

$2 5^{2} = O P^{2} + 2 4^{2}$

$625 = O P^{2} + 576$

$O P^{2} = 625 - 576$

$O P^{2} = 49$

$OP = 7 c m$

12.0Recommended Next Topics

Alternate Segment Theorem and its properties
Area Related to Circles (Sectors, Segments, and Arcs)
Inscribed Quadrilaterals (Cyclic properties and proofs)
Distance Formula and Coordinate Circles fundamentals

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Tangent in Mathematics

Master Circle Tangents & Theorems in Minutes: Learn how a straight line interacts with a circle at exactly one point. Master the fundamental properties of the point of contact, understand the two major board exam theorems, and solve high-yield geometric problems easily—all formatted in clean, plain text for simple copy-pasting.

Class: 10 Mathematics (CBSE)
Chapter: Circles
Estimated Learning Time: 15–20 Minutes

1.0Learning Outcomes

After completing this lesson, you will be able to:

Distinguish between a Tangent, a Secant, and a Chord of a circle.
State and apply Theorem 1 (Radius is perpendicular to the tangent at the point of contact).
State and apply Theorem 2 (Tangents drawn from an external point are equal in length).
Use the Pythagorean theorem to solve numerical problems involving tangent lengths.

Tangent in Mathematics

Tangents are the most common concept of two vital topics of mathematics, geometry and trigonometry, having a different meaning and use in each one. Whether it's the sophisticated flair of a line to a curve in geometry, or how angles and ratios are brought to life in trigonometry, tangents are everywhere. So, let's explore how this simple yet profoundly insightful idea is central to everything, from circle theorems to the very structure of trigonometric identities.

2.0Tangents in Geometry

In geometry, the tangent is primarily interested in lines that are tangent to curves, such as circles, and in determining relationships between points, angles, and lengths. Tangents in geometry are used in various aspects of geometry with different meanings in different contexts, such as: