New NCERT Ganita Manjari · Class 9 Maths · Part I · Chapter 5: I'm Up and Down, and Round and Round
This chapter is about circles. It defines a circle, its chords, arcs and symmetries, and then proves twelve theorems about chords, the angles they subtend, and cyclic quadrilaterals. It also explains the circumcircle of a triangle.
From the Class 9 Maths Formula Sheet, made as per the new NCERT book Ganita Manjari · Pages 24–29 of the book.
Key points
- A circle is the set of all points in a plane at the same distance (the radius) from a fixed point (the centre).
- A chord joins two points of a circle; a diameter is a chord through the centre and is the longest chord (d = 2r).
- A circle has complete rotational symmetry, and every diameter is a line of reflection symmetry.
- The locus of points equidistant from A and B is the perpendicular bisector of AB.
- Through 2 points pass infinitely many circles; through 3 collinear points none; through 3 non-collinear points exactly one (Theorem 1).
- The circumcentre of a triangle is where the perpendicular bisectors of its sides meet: inside for an acute triangle, outside for an obtuse one, and at the midpoint of the hypotenuse for a right triangle.
- Equal chords subtend equal angles at the centre, and the converse is true (Theorems 2 and 3).
- The line from the centre to the midpoint of a chord is perpendicular to it, and the perpendicular from the centre bisects the chord (Theorems 4 and 5).
- Equal chords are equidistant from the centre and conversely; a longer chord is nearer the centre (Theorems 6–8).
- The angle at the centre is double the angle on the remaining part of the circle (Theorem 9); angles in the same segment are equal, and the angle in a semicircle is 90°.
- Points on the same circle are concyclic; a quadrilateral with concyclic vertices is a cyclic quadrilateral.
- Opposite angles of a cyclic quadrilateral add up to 180°, and if they do, the quadrilateral is cyclic (Theorems 11 and 12).
Formulas
| Diameter | d = 2r |
| Chord and distance from centre | r² = d² + (½ chord)² d = perpendicular distance of the chord from the centre |
| Chord length | chord = 2√(r² − d²) |
| Smallest circle through A and B | radius = AB/2 centre = midpoint of AB |
| Theorem 9 | ∠ACB = 2∠ADB angle at centre = 2 × angle on the circle |
| Angle in a semicircle | = 90° |
| Minor and major arc angles | minor + major angle = 360° < 180° minor, > 180° major, = 180° semicircle |
| Cyclic quadrilateral | ∠A + ∠C = 180°, ∠B + ∠D = 180° |
| Exterior angle of a cyclic quadrilateral | exterior angle = interior opposite angle |
Diagrams to learn
- Circle showing chord, diameter, radius, arc and angle at the centre
- Lines of symmetry of a circle, square, regular pentagon and hexagon
- Circles through two points with centres on the perpendicular bisector
- Circumcircles of acute, obtuse and right triangles
- Perpendicular from the centre to a chord
- Angle at the centre is double the angle on the circle (two cases)
- Angles in the same segment and angle in a semicircle
- Cyclic quadrilateral
- Only two perpendicular bisectors are needed to find the circumcentre; the third one automatically passes through the same point (use it as a check).
- The triangle formed by a chord and the centre is always isosceles (two sides are radii), so its base angles are equal.
- Only points on the circle see a chord at the same angle: a point inside sees a bigger angle, a point outside a smaller one.
- The only cyclic parallelogram is a rectangle. A rhombus that is not a square is never cyclic.
A real page from the book
Get the full chapter in the book
This page is a summary. The Class 9 Maths Formula Sheet has the complete chapter in colour, with every diagram and worked example.
Frequently asked questions
What is the angle in a semicircle?
The angle in a semicircle is 90°. If AB is a diameter, the angle at the centre is 180°, so the angle at any point on the circle is half of it, 90°.
What is the property of a cyclic quadrilateral?
The opposite angles of a cyclic quadrilateral add up to 180°: ∠A + ∠C = 180° and ∠B + ∠D = 180°. Conversely, if two opposite angles add up to 180°, the quadrilateral is cyclic.
How do you find the length of a chord?
If r is the radius and d is the perpendicular distance of the chord from the centre, then chord length = 2√(r² − d²). For r = 10 cm and d = 6 cm, the chord is 16 cm.
← Part I · Chapter 4: Exploring Algebraic IdentitiesPart I · Chapter 6: Measuring Space: Perimeter and Area →
