Chapter 10 — Circles

A circle is the set of all points in a plane that are at a fixed distance from a fixed point. The fixed point is called the centre and the fixed distance is called the radius.

Key terms and their meanings:

• Centre (O): The fixed point from which all points on the circle are equidistant.
• Radius (r): The distance from the centre to any point on the circle. Every radius has the same length.
• Diameter (d): A line segment passing through the centre whose endpoints lie on the circle. d = 2r.
• Circumference: The length or perimeter of the circle (complete boundary).
• Chord: A line segment with both endpoints on the circle. A diameter is a chord that passes through the centre.
• Arc: A continuous part of the circle's circumference between two points. A minor arc is the shorter arc between the points, a major arc is the longer one.
• Semicircle: An arc equal to half the circumference (endpoints of a diameter).
• Quadrant: One fourth of a circle (90° sector).
• Sector: The region bounded by two radii and the included arc (like a pizza slice). Its size is measured by the central angle.
• Segment: The region bounded by a chord and the corresponding arc (area of sector minus area of triangle formed by the radii).
• Tangent: A line that touches the circle at exactly one point. The radius drawn to the point of contact is perpendicular to the tangent.
• Secant: A line that intersects the circle at two points.
• Concentric circles: Two or more circles that share the same centre but have different radii.
• Central angle: An angle whose vertex is at the centre and whose sides are radii; it subtends an arc.

Simple properties to remember (useful for later proofs):

• The perpendicular from the centre to a chord bisects the chord and its arcs.
• Equal chords of a circle subtend equal arcs and are equidistant from the centre.
• A tangent is perpendicular to the radius at the point of contact.

Definition: Concentric circles are two or more circles that have the same centre but different radii. They lie one inside another and do not intersect.

Key properties:

• All concentric circles share a common centre O. If radii are R and r (R > r), the circles are represented by (x - h)^2 + (y - k)^2 = R^2 and (x - h)^2 + (y - k)^2 = r^2.
• Concentric circles never meet (no common point) when radii are different.
• There is no common tangent line that touches two distinct concentric circles at the same point; in fact no single straight line can be tangent to both circles simultaneously.
• If two concentric circles subtend the same central angle θ, lengths of their corresponding arcs are proportional to their radii (arc length = rθ), and areas of corresponding sectors are proportional to the square of the radii (sector area = 1/2 r^2 θ).
• The region between two concentric circles is called an annulus (a ring). Its area equals the difference of the areas of the larger and smaller circles: π(R^2 − r^2).

Geometric consequences and uses: Because all radii from the centre to the circles are collinear for a given direction, the shortest distance between the two circles along a ray from the centre is |R − r|. Concentric circles appear often in problems that compare lengths or areas of arcs, sectors and rings.

Algebraic form / family of circles: For centre (h,k), the family of concentric circles is (x − h)^2 + (y − k)^2 = r^2, with r varying. In the plane, concentric circles can be used to model levels of equal distance from a point (level curves).

Definition: A central angle of a circle is an angle whose vertex is the centre of the circle and whose sides (arms) are radii. If O is the centre and A, B are points on the circle, then ∠AOB is a central angle.

A inscribed angle (or angle subtended at the circumference) is an angle whose vertex lies on the circle and whose sides pass through two other points on the circle. If C is on the circle and A, B are two other points on the circle, then ∠ACB is an inscribed angle subtending arc AB.

Key relationship (central–inscribed angle theorem): An inscribed angle is half the central angle subtending the same arc. In symbols, if ∠AOB is the central angle and ∠ACB is an inscribed angle subtending the same arc AB, then ∠ACB = (1/2) ∠AOB.

Reason (sketch of proof): Join OA, OB and OC (radii). Triangles OCA and OCB are isosceles (OA = OC = OB). Using base angles in these isosceles triangles and angle-chasing, one gets ∠AOB = 2·∠ACB. Thus the inscribed angle equals half the central angle subtending the same arc.

Other important results:

• Angles subtending the same arc (or chord) at the circumference are equal. That is, if points C and D lie on the same arc AB, then ∠ACB = ∠ADB.
• Angle in a semicircle is a right angle (Thales' theorem): an angle subtended by a diameter is 90°.
• Opposite angles of a cyclic quadrilateral are supplementary: for a quadrilateral ABCD inscribed in a circle, ∠A + ∠C = 180° and ∠B + ∠D = 180°.

These properties are widely used in solving geometry problems involving circles, construction of angles, and proving equalities or perpendicularities.

What is a chord? A chord of a circle is a line segment whose endpoints lie on the circle. A diameter is a chord passing through the centre and is the longest possible chord.

Basic definitions

• Centre: the fixed point equidistant from all points on the circle.
• Radius (r): distance from centre to any point on circle.
• Chord: segment joining two points on the circle.
• Perpendicular distance (d) from centre to a chord: shortest distance (a perpendicular).

Key properties and theorems (with short justifications)

• Perpendicular from centre to a chord bisects the chord. If O is centre and AB a chord, drop OM perpendicular to AB. Triangles OMA and OMB are congruent (OA = OB = r, OM common, right angles), so AM = MB.
• Line through centre that bisects a chord is perpendicular to it (converse). If a line through O bisects AB at M, then OA = OB and OM is common; using triangles again you get right angles at M.
• Equal chords subtend equal central angles. If chords AB and CD are equal, triangles OAB and OCD (O = centre) are congruent so ∠AOB = ∠COD.
• Equal chords are equidistant from the centre (and converse). If AB = CD then the perpendicular distances from O to AB and CD are equal. Conversely, chords at equal distances from O are equal in length.
• Diameter perpendicular to a chord bisects the chord and its arcs. This is a special case of the first property where the line through O is a diameter; it bisects the chord and the two arcs cut by the chord.

Relation between chord length, radius and distance from centre

For a chord of length L at distance d from the centre in a circle of radius r, use the right triangle made by half the chord: (L/2)^2 + d^2 = r^2. Hence

L = 2 sqrt(r^2 - d^2)

Why these matter

These theorems let you construct circles and chords, solve for lengths and angles, and reason about symmetry. They are widely used in geometric constructions, proofs and problem solving.

Overview: These are basic theorems about angles made by chords and arcs in a circle. They help in identifying right angles and equal inscribed angles.

1. Angle in a Semicircle (Thales' Theorem): If AB is a diameter of a circle and C is any point on the circle (other than A or B), then ∠ACB = 90° (i.e., the angle subtended by a diameter at the circumference is a right angle).

Short proof (coordinate/vector): Put the circle center at the origin with radius r. Let A(−r,0), B(r,0) and C(x,y) such that x² + y² = r². Vectors CA = (−r−x, −y) and CB = (r−x, −y). Their dot product is 0, so CA ∘ CB = 0 ⇒ ∠ACB = 90°.

2. Angles in the Same Segment: If chord AB subtends angles ∠ACB and ∠ADB at two points C and D on the same arc (same side of AB), then ∠ACB = ∠ADB. In words: angles in the same segment of a circle are equal.

Reason (using center O): Let O be the centre. The central angle AOB subtending chord AB equals twice any angle subtended by AB at the circumference: ∠AOB = 2×∠ACB and ∠AOB = 2×∠ADB. Hence ∠ACB = ∠ADB.

Related facts (useful):

• Angle subtended by a chord at the center = 2 × angle subtended by the same chord at circumference on the same side.
• Angles subtended by the same chord on the same side are equal.
• Opposite angles of a cyclic quadrilateral are supplementary (sum to 180°).

How to apply: To prove a triangle is right-angled, show one side is a diameter. To prove two inscribed angles equal, show they subtend the same chord and lie on the same side of it.

Definition: A tangent to a circle is a straight line that touches the circle at exactly one point. The point where the tangent touches the circle is called the point of contact or point of tangency.

Basic ideas:

• A tangent meets the circle in exactly one point.
• A line that intersects a circle in two distinct points is a secant; one that does not meet it at all is an external line.

Important properties (with short justifications):

• Radius–tangent perpendicularity: The radius drawn to the point of contact is perpendicular to the tangent.
  Justification (idea): If the radius to the contact point were not perpendicular, the line through that radius point would cut the circle at a second point, contradicting that the tangent meets the circle only once. Hence radius at point of contact is perpendicular to the tangent.
• Converse: A line perpendicular to a radius at its end point on the circle is a tangent.
  Idea: If a line through a point on the circle is perpendicular to the radius, it cannot meet the circle at any other point, so it is a tangent.
• Equal tangents from an external point: If two tangents are drawn to a circle from an external point P meeting the circle at A and B, then PA = PB.
  Justification (idea): OA and OB are radii. Triangles OAP and OBP are right triangles (OA and OB perpendicular to tangents). Using the equal hypotenuse/leg relationships or Pythagoras with OP common, we get PA = PB.
• Tangent length formula: If OP = d is the distance from the circle center O to an external point P and r is the radius, then length of the tangent from P to the circle is sqrt(d^2 - r^2).
  Derivation: In right triangle OAP, PA^2 = OP^2 - OA^2 = d^2 - r^2, so PA = sqrt(d^2 - r^2).
• Angle between tangent and chord (Alternate segment theorem): The angle between a tangent at point A and a chord AB through A equals the angle in the opposite arc (the angle in the alternate segment).
  Idea: This relates a tangent-chord angle to an inscribed angle subtending the same arc.
• Unique tangent at non-singular point: At any given point on a circle there is exactly one tangent line (except at points where the curve is not smooth, which does not occur for circles).

Use and interpretation: A tangent line gives the instantaneous straight-line direction that just touches the circle without cutting it. In many applications the tangent models the immediate contact direction (for example, a wheel on the road).

Short examples of reasoning you will use:

• To show two tangent segments from the same external point are equal, form right triangles with the center and apply Pythagoras.
• To construct a tangent from an external point, draw the line from the external point to the center, draw a circle with that as radius equal to the distance to the center, or use perpendicular from center to the intended tangent point.

Definition. A tangent to a circle is a line that touches the circle at exactly one point. If the point from which a tangent is drawn lies outside the circle, that line segment from the external point to the point of contact is called a tangent segment (an external tangent segment).

Key results (Class 9 level).

• From an external point P to a circle with centre O and radius r, two tangents PA and PB can be drawn. These tangent segments are equal: PA = PB.
• If OP is the distance from the centre O to the external point P, then the square of the length of the tangent equals the difference of squares of OP and r (Pythagorean relation):
  PA^2 = OP^2 − r^2.

Why this is true (proof idea). Join O to A (point of contact). OA = r and OA ⟂ PA (radius to point of contact is perpendicular to tangent). In triangle OAP, angle at A is 90°, so by Pythagoras OP^2 = OA^2 + AP^2, i.e. AP^2 = OP^2 − r^2. For two tangents from P, triangles OAP and OBP are congruent, giving PA = PB.

Common tangents to two circles. For two circles with centres C1, C2, radii r1, r2 and centre distance d = C1C2, there are two types of common tangents:

• Direct (external) common tangents — the tangent line touches both circles on the same side of the line joining the centres. The length of the segment between the two points of contact on the tangent is
  L_direct = sqrt(d^2 − (r1 − r2)^2),
  provided the expression under the square root is non‑negative (i.e. the circles are not one strictly inside the other).
• Transverse (internal) common tangents — the tangent line crosses between the circles; the length of the segment between points of contact is
  L_transverse = sqrt(d^2 − (r1 + r2)^2),
  which exists only when d ≥ r1 + r2 (circles separate enough).

Geometric idea for these formulas. Draw radii to the points of contact; each radius is perpendicular to the tangent. This gives right triangles whose legs include the radii and half (or the whole) of the tangent-segment length; apply Pythagoras to relate d, r1, r2 and the tangent length.

Conditions (quick summary).

• If d > r1 + r2: the circles are separate — there are 4 common tangents (2 direct, 2 transverse).
• If d = r1 + r2: circles touch externally — tangent lengths reduce accordingly (one transverse tangent length becomes 0).
• If |r1 − r2| < d < r1 + r2: the circles intersect — only 2 direct tangents exist.
• If d = |r1 − r2|: circles touch internally — exactly 1 common tangent.
• If d < |r1 − r2|: one circle lies inside the other — no common tangents.

Applications: These results are used in construction problems, in optimizing mechanical contacts (pulleys, gears), in optics (tangent rays), and in many 2D geometry problems.

Definitions

Circle: set of all points in a plane at a fixed distance (radius r) from a fixed point (center O).

Arc: a continuous part of the circumference between two points A and B. The shorter arc between A and B is the minor arc; the longer is the major arc. An arc is usually specified by its central angle θ (angle AOB).

Sector: the region enclosed by two radii OA and OB and the arc AB. The region with the smaller central angle is the minor sector, the other is the major sector. A sector is like a "slice" of the circle.

Segment: the region bounded by a chord AB and the arc AB. The region bounded by the chord and the smaller arc is the minor segment; with the larger arc is the major segment. A segment = sector minus the isosceles triangle formed by the two radii.

Relations and key ideas

• The central angle θ (in degrees or radians) determines arc length and sector area: arc length is proportional to θ; sector area is proportional to θ.
• Radians provide the simplest formulas: 1 full circle = 2π radians; to convert degrees to radians multiply by π/180 (θ_rad = θ_deg * π/180).
• Chord length is related to central angle: c = 2r sin(θ/2) (θ in radians or degrees as long as sine uses same-angle unit).
• Area of the isosceles triangle OAB formed by the two radii = (1/2) r^2 sin θ (θ in radians or degrees—using sine accordingly).
• Area of a segment = area of corresponding sector − area of triangle OAB. For the major segment, subtract the minor segment from the whole circle area.
• Perimeter of a sector = arc length + two radii = (arc length) + 2r. Perimeter of a segment = arc length + chord length.

Units and tips

• Keep angle units consistent when using formulas (radians simplify many formulas).
• When θ is given in degrees, use the fraction θ/360 for parts of the full circle.

Overview: "Constructions related to circles" covers methods to draw tangents and circles under given conditions using ruler and compass, and the geometric reasoning behind them. Key ideas used are: the radius to a point of contact is perpendicular to the tangent, and Thales' theorem (angle in a semicircle is a right angle).

Basic constructions and reasoning:

1. To draw a circle when the centre and radius are given — Place the compass at the centre, open it to the given radius and draw the circle. (This is the basic starting construction.)
2. Tangent to a circle at a given point on the circle
   1. Given circle with centre O and point A on the circle. Join O to A.
   2. At A, draw a line perpendicular to OA. That line is the tangent to the circle at A.
   3. Reason: OA is a radius and the radius at the point of contact is perpendicular to the tangent.
3. Tangents from an external point to a circle (construct both tangents)
   1. Let O be the centre of the circle and P be a point outside the circle.
   2. Join OP. Construct the circle having OP as diameter (i.e., with centre M, the midpoint of OP, and radius = OP/2).
   3. The circle with diameter OP intersects the given circle at the points of contact T₁ and T₂ (if any). Join P to each intersection; PT₁ and PT₂ are the required tangents.
   4. Reason: For a tangent PT, angle OTP is 90°. By Thales' theorem any point T on the circle with diameter OP satisfies ∠OTP = 90°, so intersection points give right angles and thus tangency.
4. Circle through three non-collinear points (circumcircle)
   1. Given three non-collinear points A, B, C. Draw segments AB and BC.
   2. Construct perpendicular bisectors of AB and BC. Their intersection O is the centre of the circle passing through A, B and C.
   3. With centre O and radius OA draw the circle — it passes through A, B and C.
   4. Reason: Points equidistant from A and B lie on the perpendicular bisector of AB. Intersection of two such bisectors gives a point equidistant from A, B and C — the centre.
5. Length of tangent from an external point (useful relation)

   If P is outside the circle with centre O and radius r, and PT is the tangent length from P to the point of contact T, then PT = sqrt(OP^2 − r^2). This follows from right triangle OTP where OT = r and ∠OT P = 90°.

Practical tips when constructing:

• Always mark the centre and given points clearly and use a sharp pencil for compass arcs.
• To find the midpoint of a segment, draw equal arcs from endpoints and join their intersection points — then draw the perpendicular bisector.
• When constructing tangents from P, if OP < r then no real tangent exists (point is inside the circle); if OP = r exactly one tangent exists (point on the circle).

Connections to proofs and problems: These constructions are used to prove properties (radius ⟂ tangent, equal tangents from an external point) and to solve geometric construction problems such as drawing a circle that touches given lines or passes through given points.

Overview
In the Class 9 chapter on Circles, many questions ask you to prove relationships between angles, chords, radii and tangents. Typical problems require applying a few basic circle theorems together with elementary triangle congruence and isosceles-triangle arguments.

Common problem types

• Angle relations: show an angle subtended by a chord at the centre is twice the angle subtended at the circumference, angle in a semicircle is 90°, angles in the same segment are equal.
• Chord and distance relations: prove a perpendicular from the centre bisects a chord, prove equal chords are equidistant from the centre and vice versa.
• Tangent problems: show a tangent is perpendicular to the radius at the point of contact, prove lengths of tangents from an external point are equal, relate tangent–chord angle to angle in the opposite arc.
• Proofs combining the above: e.g. prove a quadrilateral is cyclic or prove right angles using chord/arc properties.

Proof techniques and strategy

• Draw a clear diagram and label given elements (centre O, chord AB, point P etc.).
• Mark equal lengths and parallel/perpendicular lines as you infer them.
• Use congruent triangles (SSS, SAS, ASA, RHS) to prove equality of angles or lengths. Many circle proofs reduce to showing two triangles are congruent.
• Exploit isosceles triangles: if OA = OB (radii), triangle OAB is isosceles so base angles are equal.
• Apply basic circle theorems: central angle = 2 × inscribed angle; angle in a semicircle = 90°; perpendicular from centre bisects chord; equal chords subtend equal angles; tangent ⟂ radius at contact.
• If direct approach is messy, use proof by contradiction: assume the negation and derive an impossible result.
• Write a short, logical sequence: what is given → which theorem/triangle used → conclusion.

Tips

• Always refer to points precisely (e.g. ∠AOB, chord AB, arc ACB) so your reasoning is unambiguous.
• When you claim triangles are congruent, explicitly state which congruence criterion you use and which corresponding parts match.
• Practice sketching accurate diagrams — many mistakes come from incorrect or incomplete figures.