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I’m Up and Down, and Round and Round – Class 9 Maths Chapter 5 Notes & NCERT PDF

Class 9MathematicsChapter 5NCERT book: Ganita Manjari - Mathematics Class 9

Notes and a simple summary of Chapter 5 of the NCERT Class 9 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 5: I’m Up and Down, and Round and Round

Introduction

Circles are everywhere in nature — ripples when raindrops hit water, the cross-section of a plant stem, a sunflower head, the full moon. Even the Gudahandi cave paintings in Odisha show circles and other geometric shapes. This chapter turns our everyday observation that every circle has a centre equally far from all its points into a definition, and then uses it to discover a chain of beautiful theorems.

A circle is the set (locus) of all points in a plane that are equidistant from a given point, the centre. The common distance is the radius. A segment joining two points of the circle is a chord; a chord through the centre is a diameter.

Key Ideas

Symmetries of a circle (5.2)

  • Complete rotational symmetry: rotate a circle about its centre by any angle and it looks the same (like a turning wheel).
  • Reflection symmetry: every diameter is a line of symmetry — fold a paper circle so the edges overlap and the crease passes through the centre.

    • How many circles? (5.3)

  • The locus of points equidistant from A and B is the perpendicular bisector of AB. So infinitely many circles pass through two points; their centres all lie on that perpendicular bisector. The smallest has AB as diameter.
  • Theorem 1: Exactly one circle passes through three non-collinear points. Its centre is where the perpendicular bisectors of AB and AC meet. No circle passes through three collinear points.
  • This circle is the circumcircle of triangle ABC and its centre is the circumcentre: inside an acute triangle, outside an obtuse triangle, and at the midpoint of the hypotenuse of a right triangle.

    • Chords and angles at the centre (5.4)

  • Theorem 2: Equal chords subtend equal angles at the centre (SSS congruence of the two centre triangles).
  • Theorem 3 (converse): Chords subtending equal angles at the centre are equal (SAS congruence).

    • Midpoints and perpendiculars (5.5)

  • Theorem 4: The line from the centre to the midpoint of a chord is perpendicular to the chord (SAS, and two equal angles on a line are each 90°).
  • Theorem 5: The perpendicular from the centre to a chord bisects it.

    • Distance of chords from the centre (5.6)

  • Theorem 6: Equal chords are equidistant from the centre (SSS or RHS congruence).
  • Theorem 7: Chords equidistant from the centre are equal.
  • Theorem 8: The longer chord is closer to the centre (Baudhāyana–Pythagoras: CF² + AF² = CG² + GD²). The diameter, at distance 0, is the longest chord.
  • Useful formula: a chord at distance d from the centre of a circle of radius r has length 2√(r² − d²).

    • Angles subtended by an arc (5.7)

  • An arc's two parts: the minor arc (angle at centre < 180°) and major arc (> 180°).
  • Theorem 9: The angle an arc subtends at the centre is double the angle it subtends at any point on the remaining part of the circle (proved with isosceles triangles and the exterior angle theorem, in two cases).
  • So all angles subtended by the same arc at points on the rest of the circle are equal.
  • Corollary: The angle in a semicircle (subtended by a diameter) is 90°.

    • Concyclic points and cyclic 4-gons (5.8)

  • Theorem 10: If segment AB subtends equal angles at C and D on the same side of AB, then A, B, C, D are concyclic.
  • Theorem 11: In a cyclic quadrilateral, opposite angles add up to 180°.
  • Theorem 12 (converse): If a pair of opposite angles of a quadrilateral add up to 180°, it is cyclic.

    • Worked Examples

  • Chord length: radius 13 cm, chord 5 cm from the centre. Half-chord = √(13² − 5²) = √144 = 12, so the chord is 24 cm.
  • Distance from centre: diameter 26 cm (r = 13), chord 24 cm. Distance = √(13² − 12²) = 5 cm.
  • Radius from a chord: chord 16 cm at distance 6 cm. r = √(8² + 6²) = 10 cm.
  • Parallel chords: chords 6 cm and 8 cm on opposite sides of the centre of a circle of radius 5 cm. Distances are √(25 − 9) = 4 cm and √(25 − 16) = 3 cm, so the midpoints are 7 cm apart.
  • Arc angle: an arc subtends 70° at the centre, so it subtends 35° at any point on the rest of the circle.
  • Cyclic quadrilateral: ∠P = (2x + 10)°, ∠R = (3x − 20)°, opposite angles. 5x − 10 = 180 → x = 38, so ∠P = 86°, ∠R = 94°.

    • Figure it Out — Hints

  • Always draw the figure, mark equal radii, and look for congruent triangles (SSS, SAS, RHS).
  • For chord–distance–radius problems draw the perpendicular from the centre: it halves the chord and makes a right triangle.
  • A quadrilateral with angles 80°, 110°, 100°, 70° can't be cyclic in that order — check whether opposite angles sum to 180°.
  • To find the centre of a paper circle: draw two chords and their perpendicular bisectors; they meet at the centre.

    • Maths Words

  • Circle (वृत्त) · Centre (केंद्र) · Radius (त्रिज्या) · Diameter (व्यास) · Chord (जीवा) · Arc (चाप)
  • Locus (बिंदुपथ) · Perpendicular bisector (लंब समद्विभाजक) · Circumcircle (परिवृत्त) · Circumcentre (परिकेंद्र)
  • Concyclic (एकवृत्तीय) · Cyclic quadrilateral (चक्रीय चतुर्भुज) · Subtend (अंतरित करना) · Collinear (संरेख)

    • Historical / Indian Connections

  • The Baudhāyana–Pythagoras theorem is the key tool for chord distances (Theorem 8 and the chord-length formula).
  • Early circular patterns in the Gudahandi cave paintings (Odisha) show how long humans have been drawn to this shape.

    • Common Mistakes

  • Thinking a circle can pass through three collinear points — it cannot.
  • Assuming the circumcentre is always inside the triangle.
  • Forgetting that the perpendicular from the centre bisects the chord, so using the full chord instead of half in Pythagoras.
  • Mixing up which arc is meant; the angle at the circle is half the angle of the arc not containing that point.
  • Saying all opposite angles of any quadrilateral add to 180° — only cyclic ones do.

    • Maths in Daily Life

    Wheels, gears and clock faces rely on rotational symmetry; locating the centre of a round table-top, a well or a broken plate uses perpendicular bisectors; engineers use chord–radius relations to design arches and bridges.

    💡 Key Learning Points

    • ✓A circle is the locus of points equidistant from a centre; it has complete rotational symmetry and every diameter is a line of symmetry.
    • ✓Infinitely many circles pass through two points, exactly one through three non-collinear points (the circumcircle), none through three collinear points.
    • ✓Equal chords ⇔ equal angles at the centre ⇔ equal distances from the centre; the perpendicular from the centre bisects a chord.
    • ✓The angle an arc makes at the centre is twice the angle at any point on the rest of the circle; the angle in a semicircle is 90°.
    • ✓Opposite angles of a cyclic quadrilateral add to 180°, and the converse identifies cyclic quadrilaterals.

    👨‍🏫 Teaching Tips

    • →Begin with paper-circle folding: find diameters as creases and locate the centre by intersecting two perpendicular bisectors.
    • →Use the 'rotating wheel with a thread chord' model to motivate Theorems 2 and 3.
    • →Let students draw acute, obtuse and right triangles with their circumcircles and classify where the circumcentre falls.
    • →Record chord length vs distance from centre in a class table before proving Theorem 8.
    • →Use GeoGebra or a protractor activity: measure angles subtended by one arc at several points on the circle.
    • →Stress 'Given / To show / Why is this true' structure for every proof, as the book does.
    • →Give the 80°, 110°, 100°, 70° quadrilateral as a discussion question on cyclic quadrilaterals.

    📋 Assessment Questions

    1. Find the chord length when radius and distance from centre are known, and vice versa.
    2. Explain why exactly one circle passes through three non-collinear points.
    3. Prove that the perpendicular from the centre bisects a chord.
    4. Find unknown angles using the arc-angle theorem and the cyclic quadrilateral property.
    5. Show that a rectangle is the only parallelogram that can be inscribed in a circle.

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    Keep learning

    All chapters in Ganita Manjari - Mathematics Class 9

    1. 1. Orienting Yourself: The Use of Coordinates
    2. 2. Introduction to Linear Polynomials
    3. 3. The World of Numbers
    4. 4. Exploring Algebraic Identities
    5. 5. I’m Up and Down, and Round and Round
    6. 6. Measuring Space: Perimeter and Area
    7. 7. The Mathematics of Maybe: Introduction to Probability
    8. 8. Predicting What Comes Next: Exploring Sequences and Progressions