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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.

5.7 ANGLES SUBTENDED BY AN ARC

An arc is a connected portion of a circle defined by two endpoints. There are two arcs between any two points: a minor arc (smaller) and a major arc (larger). The angle subtended by an arc at the centre is the angle formed by the radii joining the centre to the arc's endpoints. Theorem 9 states that the angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the circle outside the arc. This is proved using properties of isosceles triangles and the exterior angle theorem. A corollary is that the angle subtended by a diameter at any point on the circle is 90°. Additionally, all points on the circle outside the arc subtend equal angles to that arc, a unique property of circles. These results are fundamental in understanding circle geometry and have many applications.

📊 Diagram: Fig. 5.17 and 5.18 illustrate major and minor arcs. Fig. 5.19 shows arcs with angles at centre. Fig. 5.20 to 5.24 provide detailed geometric proofs and corollaries related to angles subtended by arcs.

🧪 Activity: Draw a circle and chord AB; measure angles subtended by an arc at the centre and at various points on the circle outside the arc to observe the relationship.

🔗 Connection: These angle properties lead to the study of concyclicity and properties of cyclic quadrilaterals.

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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