10.1
Angle subtended by a chord
Longer chords subtend larger angles at the centre. Equal chords of a circle subtend equal angles at the centre — and the converse is true too.
Theorems 10.1 & 10.2
01 — Concepts
What Chapter 10 covers
Six sections, in exactly the order your NCERT textbook presents them.
10.2
Perpendicular from the centre
The perpendicular drawn from the centre to a chord bisects the chord; and the line from the centre to the midpoint of a chord is perpendicular to it.
Theorems 10.3 & 10.410.3
Circle through three points
There is exactly one circle through three non-collinear points — its centre is where the perpendicular bisectors of two of the segments meet.
Theorem 10.510.4
Equal chords & their distances
Equal chords are equidistant from the centre, and chords equidistant from the centre are equal. Of two unequal chords, the longer is nearer the centre.
Theorems 10.6 & 10.710.5
Angle subtended by an arc
The angle subtended by an arc at the centre is double the angle it subtends at any point on the remaining part of the circle. Angles in the same segment are equal.
Theorems 10.8 & 10.910.6
Cyclic quadrilaterals
A quadrilateral with all four vertices on a circle is cyclic. The sum of either pair of opposite angles of a cyclic quadrilateral is 180° — and the converse holds.
Theorems 10.10–10.12
02 — Key terms
Learn these definitions verbatim
One-mark questions in the exam are almost always lifted straight from this list.
01
Circle
The collection of all points in a plane which are at a fixed distance from a fixed point in the plane.
02
Centre
The fixed point, written as O. All radii of a circle are equal in length.
03
Radius
The fixed distance from the centre to any point on the circle; also the line segment joining them.
04
Diameter
A chord passing through the centre. It is the longest chord of a circle and d = 2r.
05
Chord
A line segment joining any two points on the circle.
06
Arc
A connected piece of the circle. An arc whose ends are the ends of a diameter is a semicircle.
07
Sector
The region between an arc and the two radii joining the centre to the endpoints of the arc.
08
Segment
The region between a chord and either of its arcs — minor or major segment.
09
Cyclic quadrilateral
A quadrilateral with all four vertices lying on a circle; it is inscribed in the circle.
i
Remember: a circle divides its plane into three parts — the interior, the circle itself and the exterior. The circle and its interior together make the circular region.
03 — Theorems
Theorems 10.1 – 10.12
The complete list from the NCERT chapter, with the congruence test each proof uses.
Equal chords, equal angles
Equal chords of a circle subtend equal angles at the centre.
SSS congruence
Converse of 10.1
If the chords of a circle subtend equal angles at the centre, then the chords are equal.
Converse
Perpendicular bisects the chord
The perpendicular from the centre of a circle to a chord bisects the chord.
RHS congruence
Line through the centre
The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
Converse
One circle through three points
There is one and only one circle passing through three given non-collinear points.
Perp. bisectors
Equal chords, equal distance
Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).
RHS congruence
Equidistant chords are equal
Chords equidistant from the centre of a circle are equal in length.
Converse
Arc at centre vs on circle
The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Exterior angleMost asked
Angles in the same segment
Angles in the same segment of a circle are equal. The angle in a semicircle is a right angle.
From 10.8
Four points on a circle
If a line segment joining two points subtends equal angles at two other points lying on the same side, the four points lie on a circle.
Contradiction
Cyclic quadrilateral
The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
From 10.8
Converse of 10.11
If the sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.
Converse
04 — Formulas
Formulas you must remember
Use π ≈ 22/7 or 3.14 unless the question says otherwise.
Diameterd = 2rpasses through the centre
CircumferenceC = 2πrlength around the circle
AreaA = πr²area of the circular region
Sector area(θ/360°) × πr²θ = central angle
Arc length(θ/360°) × 2πrθ = central angle
CircumradiusR = c/2right triangle, c = hypotenuse
Worked example
NCERT Exercise 10.6 — two parallel chords
The lengths of two parallel chords AB and CD of a circle are 6 cm and 8 cm. If the smaller chord AB is at a distance 4 cm from the centre, find the distance of CD from the centre.
- Draw OM ⊥ AB. By Theorem 10.3 M is the midpoint of AB, so AM = 3 cm.
- In right ΔOMA: r = OA = √(OM² + AM²) = √(16 + 9) = 5 cm.
- Draw ON ⊥ CD. By Theorem 10.3 again, N is the midpoint of CD, so CN = 4 cm.
- In right ΔONC: ON = √(OC² − CN²) = √(25 − 16) = 3 cm.
Note how the longer chord ends up nearer the centre — exactly what Theorem 10.6 predicts.
05 — Solutions
NCERT exercises, solved
Open an exercise to see the full solution, written the way the textbook expects it.
Exercise 10.1 — Fill in the blanks Warm-up
(i) The centre of a circle lies in the interior of the circle. (ii) A point whose distance from the centre is greater than its radius lies in the exterior. (iii) The longest chord of a circle is a diameter. (iv) An arc whose ends are the ends of a diameter is a semicircle. (v) Segment of a circle is the region between an arc and the chord of the circle. (vi) A circle divides the plane, on which it lies, into three parts.
Exercise 10.2 — True or false Concepts
Line segment joining the centre to any point on the circle is a radius — True. A circle has only a finite number of equal chords — False (there are infinitely many). If a circle is divided into three equal arcs, each is a major arc — False (each is a minor arc). A chord of a circle which is twice as long as its radius is a diameter — True. A sector is the region between the chord and its corresponding arc — False (that is a segment).
Classic problem — chord equal to the radius Theorems 10.8, 10.11
A chord of a circle is equal to the radius. ΔAOB is equilateral (OA = OB = AB), so ∠AOB = 60°. By Theorem 10.8, the angle at any point on the major arc is 30°, and by Theorem 10.11 the angle at any point on the minor arc is 180° − 30° = 150°.
Classic problem — angle in a semicircle Theorem 10.8
If AB is a diameter and C any point on the circle, arc AB subtends 180° at the centre O. So by Theorem 10.8 the angle it subtends at C is half of that — 90°. This is exactly why the circumcentre of a right triangle is the midpoint of its hypotenuse.
Classic problem — circles on two sides of a triangle Theorem 10.9
Circles are drawn taking two sides of a triangle as diameters. Their second intersection point B makes ∠ADB = 90° = ∠ACB (angles in a semicircle), so by Theorem 10.10 the four points lie on one circle — B lies on the third side.
06 — FAQ
Exam questions, answered
The five things students ask most before the Class 9 circles test.
Which theorem is asked most often?
Theorem 10.8 — the arc at the centre is double the angle on the circle. It hides inside most “find ∠BCD” problems and inside every cyclic-quadrilateral question too.
Is Chapter 10 hard?
Not if you separate the two halves: sections 10.1–10.4 are all about chords and congruent triangles; 10.5–10.6 are all about angles. Almost every proof is SSS or RHS on two triangles — once you spot the triangles, the marks follow.
Do I need to memorise the proofs?
Yes — CBSE regularly asks the proof of Theorems 10.3, 10.5, 10.8 and 10.11 in exams. They are short (four to six lines) and each reuses the same two congruence rules.
Why do cyclic quadrilaterals add to 180°?
Each pair of opposite angles takes half of two arcs that together make the full circle: ½(360°) = 180°. It is just Theorem 10.8 applied twice.
How much of the Class 10 syllabus does this chapter hold up?
A lot — tangents (Class 10, Chapter 10) restart from this chapter’s results, and constructions of circumcircles build directly on Theorem 10.5.
One chapter. Twelve theorems. Full marks.
Read the notes once, then redo every NCERT exercise without looking at the solutions.