Concept Scroll LogoConceptScroll
Join or Sign In
MathematicsClass 9I’m Up and Down, and

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

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

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

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

5.7 ANGLES SUBTENDED BY AN ARC

This section introduces arcs of a circle and the angles they subtend at the centre and at points on the circle. An arc is a connected portion of the circle defined by two endpoints. There are two arcs between two points: the minor arc (smaller) and the major arc (larger). The angle subtended by an arc at the centre is the angle between the radii to the endpoints, measured along the arc. The key theorem (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. The theorem is proved using properties of isosceles triangles and the exterior angle theorem. The section also discusses the special case of angles subtended by a diameter, which is always 90°, a corollary of Theorem 9. The section emphasizes the unique property that angles subtended by the same arc at points on the circle are equal, a fundamental characteristic of circles.

📊 Diagram: Fig. 5.17: Major arc AYB and minor arc AXB; Fig. 5.18; Fig. 5.19; Fig. 5.20; Fig. 5.21: Angle subtended by arc AFB; Fig. 5.22; Fig. 5.23: Angles subtended by an arc are equal; Fig. 5.24

🧪 Activity: Activity: Draw a circle and measure angles subtended by an arc at the centre and at various points on the circle outside the arc.

🔗 Connection: This section leads to the study of concyclicity of points and properties of cyclic quadrilaterals.

Frequently asked questions

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 and is isosceles.

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

Let two isosceles triangles OAB and OCD be formed by chords AB and CD respectively, with OA = OB and OC = OD (radii). If AB = CD (equal bases), then by the Side-Side-Side (SSS) congruence criterion, triangles OAB and OCD are congruent because OA = OC (radii), OB = OD (radii), and AB = CD (given).

1. Can you explain why the converse to Theorem 4 is true, i.e., why does the perpendicular from the centre of a circle to a chord of the circle bisect the chord? (Hint: Use Fig. 5.12. You are told that ∠CMA = ∠CMB = 90°. You need to show that AM = BM.)

Solution:

Given: A chord AB of a circle with centre C, and a perpendicular CM from C to AB.

To prove: CM bisects AB, i.e., AM = BM.

Proof: 1. Since CM is perpendicular to AB, ∠CMA = ∠CMB = 90°. 2. In triangles CMA and CMB:

  • CM = CM (common side)
  • CA = CB (radii of the circle)
  • ∠CMA = ∠CMB = 90°

3. By RHS congruence criterion, ΔCMA ≅ ΔCMB. 4. Therefore, corresponding parts are equal, so AM = BM.

Hence, the perpendicular from the centre of a circle to a chord bisects the chord.

2. An isosceles triangle ABC is inscribed in a circle, with AB = AC. Show that the altitude from A to BC passes through the centre of the circle.

Solution:

Given: Triangle ABC is isosceles with AB = AC, inscribed in a circle with centre O.

To prove: The altitude from A to BC passes through the centre O.

Proof: 1. Since AB = AC, triangle ABC is isosceles with vertex A. 2. The altitude from A to BC is also the median and angle bisector of angle A. 3. Let the altitude from A meet BC at D. 4. Since ABC is inscribed in the circle, points A, B, C lie on the circle. 5. The perpendicular bisector of chord BC passes through the centre O. 6. Sin

Ready to ace this chapter?

Get the full I’m Up and Down, and chapter — interactive notes, diagrams, worked solutions, polls and a free practice quiz — in the ConceptScroll app.

Open in ConceptScroll →

Study smarter with ConceptScroll

Daily NCERT-aligned reels, AI doubt solving and chapter quizzes — all free.

Start learning free
#cbse notes#class 9#mathematics#ncert

Continue reading

← All articles