CLASS 9 MATHEMATICS

I’m Up and Down, and
Round and Round

Chapter 5

Notes and Chapter Summary

Ganita Manjari | Grade 9 | Part I

Source-based educational material for Class 9 students

Introduction

Circles are commonly seen in nature. Raindrops falling on water form circular ripples, plant stems have circular cross-sections, sunflower inflorescences appear circular, and the Sun and full Moon also look like circles.

Central idea: Every point on a circle is at the same distance from its centre.

To locate the centre of a circular paper:

1. Fold it so that its boundaries overlap. The crease formed is a diameter.

2. Fold it again in another direction.

3. The point where the two creases intersect is the centre.

5.1 Definitions

A circle is the set of all points in a plane that are equidistant from a fixed point on that plane. It can also be described as the locus of all points equidistant from a fixed point.

Important terms

1. Centre: The fixed point from which all points on the circle are equidistant.

2. Radius: The distance from the centre to any point on the circle. It is usually denoted by r.

3. Chord: A line segment joining any two points on a circle.

4. Diameter: A chord that passes through the centre of the circle.

5. Angle subtended by a chord at the centre: If AB is a chord and O is the centre, then ∠AOB is the angle subtended by chord AB at the centre.

Diameter = 2r

5.2 Symmetries of a Circle

Rotational symmetry

A circle has complete rotational symmetry. It looks unchanged when rotated about its centre through any angle. Therefore, a circle has infinitely many rotational symmetries.

Reflection symmetry

Every diameter divides a circle into two matching halves. Hence, every diameter is a line of reflection symmetry. Since infinitely many diameters can be drawn, a circle has infinitely many lines of reflection symmetry.

Important locus result

The perpendicular bisector of a line segment AB is the locus of all points that are equidistant from A and B. This result is used to locate the centres of circles passing through two or three given points.

5.3 How Many Circles?

Circles passing through two points

Suppose A and B are two distinct points. Any point on the perpendicular bisector of AB is equidistant from A and B and can therefore be the centre of a circle through them.

6. Infinitely many circles can pass through two given points.

7. Their centres lie on the perpendicular bisector of AB.

8. The smallest circle has the midpoint of AB as its centre and AB as its diameter.

9. Its radius is AB/2.

10. There is no largest circle through A and B.

11. As the centre moves farther from AB, the radius increases and the circle appears less curved.

Circles passing through three points

No circle can pass through three distinct collinear points. For three non-collinear points, the following result holds.

Theorem 1: There is a unique circle passing through three non-collinear points.

12. The centre lies on the perpendicular bisector of AB because OA = OB.

13. It also lies on the perpendicular bisector of AC because OA = OC.

14. These perpendicular bisectors meet at one unique point O.

15. The circle with centre O and radius OA passes through A, B and C.

Circumcircle and circumcentre

16. The circle passing through all three vertices of a triangle is its circumcircle.

17. The centre of the circumcircle is the circumcentre.

18. The triangle is inscribed in the circle, while the circle circumscribes the triangle.

19. The circumcentre is the common intersection of the perpendicular bisectors of the sides.

5.4 Chords and the Angles They Subtend

Theorem 2: Equal chords of a circle subtend equal angles at the centre.

If AB = DE, then all four segments from the centre O to the chord endpoints are radii. Thus, triangles OAB and ODE are congruent by SSS, and ∠AOB = ∠DOE.

Theorem 3 (Converse): Chords that subtend equal angles at the centre are equal.

If ∠AOB = ∠DOE, the two pairs of radii are equal. Triangles OAB and ODE are congruent by SAS, so AB = DE.

Equal chords ⇔ Equal angles at the centre

A chord and the two radii drawn to its endpoints form an isosceles triangle.

5.5 Midpoints and Perpendicular Bisectors of Chords

Theorem 4: The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.

Let M be the midpoint of chord AB and O the centre. The two triangles formed by OM are congruent. Therefore, ∠OMA = ∠OMB. These equal angles form a linear pair, so each is 90°, and OM ⟂ AB.

Theorem 5 (Converse): The perpendicular drawn from the centre of a circle to a chord bisects the chord.

If OM ⟂ AB, right triangles OMA and OMB are congruent by RHS because OA = OB and OM is common. Hence, AM = BM.

Centre joined to midpoint of chord ⇔ Perpendicular to chord

Therefore, the perpendicular bisector of every chord passes through the centre of the circle.

5.6 Distance of Chords from the Centre

The distance of a chord from the centre is the length of the perpendicular drawn from the centre to the chord. This perpendicular meets the chord at its midpoint.

Theorem 6: Chords of a circle having the same length are at the same distance from the centre.

The perpendiculars from the centre bisect both equal chords. Their half-chords are equal, their hypotenuses are radii, and the resulting right triangles are congruent by RHS. Hence, the perpendicular distances are equal.

Theorem 7 (Converse): Chords equidistant from the centre have equal lengths.

Equal chords ⇔ Equal distances from the centre

5.6.1 Which Unequal Chord Is Farther from the Centre?

Theorem 8: Of two unequal chords of a circle, the longer chord is nearer to the centre.

For a chord of length L at perpendicular distance d from the centre of a circle of radius r, the radius, half-chord and perpendicular form a right triangle.

(L/2)² + d² = r²

L = 2√(r² − d²)

d = √(r² − (L/2)²)

r = √(d² + (L/2)²)

20. The longer chord is closer to the centre, and the shorter chord is farther away.

21. When d = 0, L = 2r; therefore, the diameter is the longest chord.

22. As a chord moves away from the centre, its length decreases.

23. At distance r, the chord reduces to a single point of length zero.

5.7 Angles Subtended by an Arc

An arc is a connected portion of a circle determined by two endpoints and the curved part joining them.

24. Minor arc: The smaller arc; it subtends an angle of less than 180° at the centre.

25. Major arc: The larger arc; it subtends an angle of more than 180° at the centre.

26. The angle subtended by an arc at the centre is the angle swept by a radius while moving between the endpoints along that chosen arc.

5.7.1 Angle Subtended by an Arc at a Point on the Circle

Theorem 9: The angle subtended by an arc at the centre is double the angle subtended by the same arc at any point on the circle outside the arc.

∠AOB = 2∠APB

∠APB = ½∠AOB

Angles in the same segment

If P, Q and R lie on the same remaining part of the circle, then the same arc AB subtends equal angles at them.

∠APB = ∠AQB = ∠ARB

Corollary: Angle in a semicircle

A diameter subtends 180° at the centre. Therefore, it subtends half of 180°, or 90°, at every point on the circle.

Angle in a semicircle = 90°

5.8 Concyclicity of Points

Points that lie on the same circle are called concyclic points.

Theorem 10: If segment AB subtends equal angles at points C and D on the same side of AB, then A, B, C and D are concyclic.

∠ACB = ∠ADB ⇒ A, B, C, D are concyclic

Cyclic quadrilateral

A quadrilateral whose four vertices lie on the same circle is called a cyclic quadrilateral or cyclic 4-gon.

Theorem 11: The sum of each pair of opposite angles of a cyclic quadrilateral is 180°.

∠A + ∠C = 180°

∠B + ∠D = 180°

Theorem 12 (Converse): If a pair of opposite angles of a quadrilateral adds to 180°, then the quadrilateral is cyclic.

In a cyclic quadrilateral, an exterior angle is equal to the interior opposite angle.

Tests for concyclicity

1. A segment joining two points subtends equal angles at the other two points on the same side of the segment.

2. A pair of opposite angles of the quadrilateral formed by the four points adds to 180°.

Chapter Summary

27. A circle is the locus of points in a plane at a fixed distance from a fixed centre.

28. Every diameter is a line of reflection symmetry, and a circle has rotational symmetry through every angle.

29. Infinitely many circles pass through two points; their centres lie on the perpendicular bisector of the joining segment.

30. Exactly one circle passes through three non-collinear points.

31. The circumcentre is the intersection of the perpendicular bisectors of a triangle’s sides.

32. Equal chords subtend equal central angles, and the converse is true.

33. A line from the centre to a chord’s midpoint is perpendicular to the chord, and the converse is true.

34. Equal chords are equidistant from the centre, and the converse is true.

35. The longer of two unequal chords is closer to the centre; the diameter is the longest chord.

36. An arc subtends twice as large an angle at the centre as at a point on the remaining circle.

37. Angles in the same segment are equal, and the angle in a semicircle is 90°.

38. A cyclic quadrilateral has supplementary opposite angles; the converse is also true.