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MathematicsClass 9I’m Up and Down, and Round and Round

I’m Up and Down, and Round and Round | Class 9 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 4 min read

I’m Up and Down, and Round and Round – this guide gives you a concise, exam-ready overview of I’m Up and Down, and Round and Round from Class 9 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

5.2 SYMMETRIES OF A CIRCLE

Circles are highly symmetrical shapes. They exhibit perfect rotational symmetry, meaning rotating a circle about its centre by any angle results in the same circle. For example, a vehicle's wheel appears the same at all times during rotation, indicating this symmetry. Additionally, a circle has infinite lines of reflection symmetry, each line passing through the centre and acting as a diameter. Folding a circular paper so that its boundaries overlap creates a crease along a diameter, demonstrating reflection symmetry. This section also prompts students to think about symmetries in other shapes like squares, pentagons, and hexagons, and to consider properties related to chords and loci of points equidistant from two points.

📊 Diagram: No specific figure is referenced here, but Fig. 5.3 is relevant for understanding diameters as lines of symmetry.

🧪 Activity: Fold a circular paper so that boundaries overlap to observe the crease as a line of symmetry (diameter).

🔗 Connection: Understanding symmetry leads to exploring how many circles can pass through given points and the role of perpendicular bisectors in circle construction.

Frequently asked questions

1. Draw ΔABC with AB = 5 cm, ∠A = 70° and ∠B = 60°. Draw the circumcircle of ΔABC. Is the centre inside or outside the triangle? 2. Draw ΔABC with AB = 5 cm, ∠A = 100°, AC = 4 cm. Draw the circumcircle of ΔABC. Is the centre inside or outside the triangle? 3. Draw ΔABC, with AB = 6 cm, BC = 7 cm and CA = 7 cm. Draw the circumcircle of ΔABC. Let the circumcentre be O. Measure OA, OB, OC. 4. What is the least possible radius of a circle through two points A and B?

1. To draw ΔABC with AB = 5 cm, ∠A = 70°, and ∠B = 60°:

  • Draw a line segment AB = 5 cm.
  • At point A, construct an angle of 70°.
  • At point B, construct an angle of 60°.
  • The intersection of these two rays is point C.
  • Draw the circumcircle by finding the perpendicular bisectors of at least two sides and locating their intersection (circumcentre).
  • Since both angles are acute and sum less than 180°, the circumcentre lies inside the triangle.

2. To draw ΔABC with AB = 5 cm, ∠A = 100°, and AC

Think, Draw and Infer 1. A, B and C are three collinear points. Can you find a point P such that PA = PB = PC? What can you say about the perpendicular bisectors of AB and BC? Draw and check. Can you show that for three collinear points A, B and C, the perpendicular bisector of AB and BC are parallel? Is it possible for a circle to pass through collinear points? Can you draw a line that cuts a given circle in three distinct points? 2. The circumcircle of a given ΔABC is drawn. Can there be other triangles congruent to ΔABC that share the same circumcircle?

1. For three collinear points A, B, and C:

  • It is not possible to find a point P such that PA = PB = PC unless all three points coincide.
  • The perpendicular bisectors of AB and BC are lines perpendicular to AB and BC at their midpoints.
  • Since A, B, and C are collinear, these perpendicular bisectors are parallel lines.
  • Because the perpendicular bisectors do not intersect, there is no unique circumcentre.
  • Hence, no circle can pass through three collinear points.
  • A line cannot cut a circl
1. Show that the triangle formed by a chord and the centre of the circle is isosceles.

Consider a circle with centre O and a chord AB. Join OA and OB. Since OA and OB are radii of the circle, OA = OB. Therefore, triangle OAB has two equal sides OA and OB, making it an isosceles triangle.

2. Show that if two such isosceles triangles (occurring in the previous question) have equal base length, they are congruent to each other.

Let the two isosceles triangles be OAB and OCD, where O is the centre of the circle, and AB and CD are chords with equal length (AB = CD). Since OA = OB and OC = OD (radii of the circle), and AB = CD (given), by the SSS congruence criterion, triangle OAB is congruent to triangle OCD.

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