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Class 10 · Maths · Chapter 10 · NCERT Class 10 Mathematics

Circles Class 10 Notes

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Chapter mind map

The whole chapter at a glance: the big idea, then each branch and what sits under it.

Circles and Tangents

A study of the geometric properties and relationships between circles, secants, and tangents as defined in NCERT Class 10 Mathematics.

  1. Line-Circle Interactions

    Three distinct possibilities for a line and circle in a plane based on common points.

    • Non-intersecting Line — The line stays entirely outside the circle and shares zero common points.
    • Secant — A line intersecting the circle at exactly two points; the internal segment is a chord.
    • Tangent — A line touching the circle at exactly one point, known as the point of contact.

  2. Fundamental Tangent Theorems

    Core geometric proofs regarding perpendicularity and external tangent lengths.

    • Radius-Tangent Perpendicularity — Theorem 10.1: The tangent is perpendicular to the radius at the point of contact (90° angle).
    • Equality of External Tangents — Theorem 10.2: Tangent segments drawn from a single external point to a circle are equal in length.
    • Angle Bisector Property — The line joining the external point to the circle's centre bisects the angle between the two tangents.

  3. Limiting Cases and Counts

    Mathematical limits and the number of possible tangents based on point location.

    • Secant to Tangent Transition — A tangent is a limiting case of a secant when the endpoints of its chord coincide.
    • Parallel Tangent Limit — There can be at most two tangents parallel to a given secant on opposite sides of the circle.
    • Point Location Rules — 0 tangents from inside, 1 from on the circle, and 2 from an external point.

  4. Angular Relationships

    Supplementary and proportional relationships between angles in tangent constructions.

    • Supplementary Angles — The angle between two tangents and the angle subtended by contact points at the centre sum to 180°.
    • Chord-Radius Angle Relation — The angle between tangents is double the angle between the chord and the radius (∠PTQ = 2∠OPQ).

  5. Geometric Applications

    Properties involving concentric circles and circumscribed quadrilaterals.

    • Concentric Circle Chord — A chord of a larger circle tangent to a smaller concentric circle is bisected at the point of contact.
    • Circumscribed Quadrilateral — For a quadrilateral ABCD circumscribing a circle, the sum of opposite sides is equal: AB + CD = AD + BC.
    • Circumscribed Parallelogram — A parallelogram circumscribing a circle must be a rhombus because adjacent sides become equal.

Chapter notes

An in-depth study of the properties of tangents to a circle, including theorems on perpendicularity at the point of contact and the equality of tangent lengths from external points.

01 · Explore

Introduction to Lines and Circles

A circle is defined as the collection of all points in a plane that are at a constant distance, called the radius, from a fixed point known as the centre.

When a straight line and a circle exist in the same plane, three distinct geometric possibilities arise. First, the line may not intersect the circle at all, meaning they share no common points. This is referred to as a non-intersecting line.

Second, the line may intersect the circle at exactly two points. In this scenario, the line is called a secant. The portion of the secant lying inside the circle is known as a chord.

Third, the line may touch the circle at exactly one point. This unique line is called a tangent. The single point where the line and the circle meet is called the point of contact. The word tangent originates from the Latin word 'tangere', meaning 'to touch'.

Type of LineCommon PointsDescription
Non-intersecting0The line stays entirely outside the circle.
Secant2The line passes through the circle.
Tangent1The line touches the circle at a single point.

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What is the maximum number of common points a tangent can have with a circle?

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NCERT reference: chapter PDF pages 1, 2.

02 · Explore

Tangent as a Limiting Case

A tangent is not just a line that touches a circle; it can be mathematically understood as a special limiting case of a secant.

Consider a secant that intersects a circle at two points, P and Q. If we rotate this line while keeping point P fixed, the second point Q moves along the circumference toward P. When point Q finally coincides with point P, the secant transforms into a tangent at point P.

Alternatively, if we draw several lines parallel to a secant, the length of the chords formed by these lines gradually decreases as we move toward the edges of the circle. At the very edge, the chord length becomes zero, and the line becomes a tangent. This demonstrates that for any given secant, there can be at most two parallel tangents (one on each side of the circle).

Transition from Secant to Tangent

  1. 1

    Secant Line

    A line intersecting the circle at two distinct points, P and Q.

  2. 2

    Movement

    Point Q moves along the arc toward point P.

  3. 3

    Coincidence

    Point Q and point P become the same single point.

  4. 4

    Tangent Line

    The line now touches the circle at only one point (P).

This sequence shows how a secant becomes a tangent when the endpoints of its corresponding chord coincide.

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How many tangents can be parallel to a given secant of a circle?

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NCERT reference: chapter PDF pages 2, 3.

03 · Explore

Perpendicularity of Tangent and Radius

Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

To prove this, consider a circle with centre O and a tangent XY touching at point P. Take any point Q on the tangent XY other than P. Since XY is a tangent, it only touches the circle at P. Therefore, every other point on XY, including Q, must lie outside the circle.

Because Q lies outside the circle, the distance OQ must be greater than the radius OP (OQ > OP). This holds true for every point on the line XY except P. In geometry, the shortest distance from a point (O) to a line (XY) is the perpendicular distance. Since OP is the shortest distance, OP must be perpendicular to XY.

This property is fundamental: the radius and the tangent are always at 90° to each other at the point of contact. The line containing the radius through the point of contact is sometimes called the 'normal' to the circle at that point.

OPQTangentRadiusOP ⊥ PQ
At the contact point P, radius OP is perpendicular to tangent PQ. O is the centre and Q lies outside the circle. Learning sketch; use the labels and stated dimensions, not measurements from the picture.

Calculating Distance from Centre

OQ² = OP² + PQ²

A tangent PQ at point P of a circle with radius 5 cm meets a line through centre O at point Q such that OQ = 12 cm. Since OP is perpendicular to PQ, we use Pythagoras Theorem: 12² = 5² + PQ². 144 = 25 + PQ². PQ² = 119. PQ = √119 cm.

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What is the 'normal' to a circle?

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NCERT reference: chapter PDF pages 3, 4.

04 · Explore

Tangents from Different Points

The number of tangents that can be drawn to a circle depends entirely on the location of the point relative to the circle.

If a point lies inside the circle, no tangent can be drawn through it. Any line passing through an internal point will inevitably intersect the circle at two points, making it a secant rather than a tangent.

If a point lies on the circle, exactly one tangent can be drawn at that point. This was established by the property that only one line can be perpendicular to the radius at its endpoint on the circumference.

If a point lies outside the circle, exactly two tangents can be drawn to the circle from that point. These two tangents touch the circle at two distinct points of contact.

The length of a tangent from an external point means the finite segment from that point to the point of contact. The whole tangent line continues in both directions; it does not have a finite length.

Point LocationNumber of TangentsObservation
Inside the circle0All lines through the point are secants.
On the circle1The tangent is perpendicular to the radius at that point.
Outside the circle2Two segments of equal length can be drawn.

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How many tangents can be drawn from a point located on the circumference of a circle?

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NCERT reference: chapter PDF pages 4, 5.

The rest of this chapter

Keep reading Circles, free

  1. Locked: 1. Equality of External Tangents
  2. Locked: 2. Concentric Circles and Chords
  3. Locked: 3. Angles Between Tangents
  4. Locked: 4. Quadrilaterals Circumscribing a Circle

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