Board exam focus — Circles (CBSE & HBSE)

Theorems on chords, arcs, angles subtended and cyclic quadrilaterals. CBSE asks long-answer proofs (4-mark); HBSE often tests the inscribed angle theorem and equal-chord results. 7–9 marks.

Basic Terms & Chord Properties

Vocabulary

Theorem 1 — Equal Chords ⇔ Equal Distances

Equal chords of a circle are equidistant from the centre, and vice-versa.

Proof outline: drop perpendiculars OM, ON from centre to chords AB, CD. Right triangles OMA and ONC have OM = ON (given), OA = OC (radii), right angle ⇒ RHS ⇒ AM = CN ⇒ AB = CD.

Theorem 2 — Perpendicular from Centre Bisects the Chord

The perpendicular drawn from the centre to a chord bisects the chord. Converse: the line from the centre to the mid-point of a chord is perpendicular to it.

Theorem 3 — Unique Circle through Three Non-collinear Points

Given three non-collinear points, exactly one circle passes through all three (its centre is the circumcentre — intersection of perpendicular bisectors).

CBSE Tip: When using these theorems in proofs, write "by Theorem 9.x" or by name ("perpendicular from centre bisects chord") — never just "since chord property".

Angles Subtended by Arcs / Chords

Angle Subtended by an Arc at the Centre vs Circumference

The angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the remaining part of the circle.

Formally: arc AB subtends 2θ at the centre O ⇒ subtends θ at any point P on the major arc.

Angle in a Semicircle

The diameter subtends an angle of 90° at any point on the remaining circle. (Special case of the previous theorem with central angle = 180°.)

Equal Chords ⇔ Equal Arcs ⇔ Equal Angles at Centre

Equal chords subtend equal arcs, and equal arcs subtend equal angles at the centre. All three statements are equivalent.

Two Important Symmetries

• Angles in the same segment of a circle are equal (both subtend the same arc).
• The angle in a major segment is acute; in a minor segment is obtuse.

HBSE Tip: Mark the central angle and inscribed angle on your diagram with different colours / labels to avoid mix-ups.

Cyclic Quadrilaterals

Cyclic Quadrilateral

A quadrilateral whose all four vertices lie on the same circle.

Key Theorem

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. ∠A + ∠C = 180° and ∠B + ∠D = 180°.

Converse Theorem

If the sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic (can be inscribed in a circle).

Useful Corollary — Exterior Angle

The exterior angle of a cyclic quadrilateral = interior opposite angle.

Identifying a Cyclic Quadrilateral

A quadrilateral is cyclic iff any one of these holds:

• Opposite angles sum to 180°.
• All four vertices are concyclic (lie on a common circle).
• A side subtends equal angles at the other two vertices.

Mnemonic

All cyclic quadrilaterals have supplementary opposite angles — both pairs.

CBSE Tip: Use the converse when proving that four given points are concyclic — show that one pair of opposite angles sums to 180°.

Frequently asked questions

Are these Circles notes free?

Yes — the Circles notes for Mathematics (Class 9) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Circles notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Circles chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Circles.