I'm Up and Down, and Round and Round – Class 9 Mathematics notes, important questions and MCQs

Chapter overview

This chapter studies circles: their parts, symmetry, chords, arcs and the angles they make. Key results connect chords with their distance from the centre, the angle at the centre with the angle on the circle, and the opposite angles of a cyclic quadrilateral.

Circle basics and chords

• Radius r, diameter 2r, chord (segment joining two points on the circle) and arc (part of the circle); the diameter is the longest chord.
• Every diameter is a line of symmetry of the circle, so a circle has infinitely many lines of symmetry.
• Exactly one circle passes through three points that are not on one line; no circle passes through three collinear points.
• The perpendicular from the centre to a chord bisects the chord, and the line joining the centre to a chord's midpoint is perpendicular to it.

Equal chords and distances

• Equal chords subtend equal angles at the centre, and chords subtending equal angles at the centre are equal.
• Equal chords are equidistant from the centre; a chord nearer the centre is longer.
• For a chord at distance d from the centre: chord length = 2√(r² − d²).

Angles and cyclic quadrilaterals

• The angle subtended by an arc at the centre is double the angle it subtends at any point on the remaining part of the circle.
• Angles in the same segment are equal; the angle in a semicircle is 90°.
• In a cyclic quadrilateral, each pair of opposite angles adds up to 180°; an exterior angle equals the interior opposite angle.

Key terms

Chord

a line segment joining any two points on a circle

Arc

a part of the circle between two points on it

Concyclic points

points that lie on the same circle

Cyclic quadrilateral

a quadrilateral whose four vertices lie on a circle

Formulas

• Chord length = 2√(r² − d²)
• Angle at centre = 2 × angle on the circle (same arc)
• Angle in a semicircle = 90°
• Opposite angles of a cyclic quadrilateral: ∠A + ∠C = 180°

Important questions with answers

Quick revision

• Perpendicular from centre bisects a chord.
• Equal chords ↔ equal distances ↔ equal central angles.
• Central angle = 2 × inscribed angle; semicircle angle = 90°.
• Cyclic quadrilateral: opposite angles sum to 180°.

Exam tips

• In chord problems, draw the perpendicular from the centre; it creates a right triangle for Pythagoras.
• Mark the arc first, then identify which angle is at the centre and which is on the circle.

Practice MCQs

1. The longest chord of a circle is the

   1. radius
   2. arc
   3. diameter
   4. tangent
   Show answer

   Answer: (c) diameter. It passes through the centre.

2. The angle in a semicircle is

   1. 45°
   2. 60°
   3. 90°
   4. 180°
   Show answer

   Answer: (c) 90°. The arc is a semicircle, so the angle is half of 180°.

3. How many circles can pass through three non-collinear points

   1. none
   2. infinitely many
   3. two
   4. exactly one
   Show answer

   Answer: (d) exactly one. Their perpendicular bisectors meet at one centre.

4. A chord is 3 cm from the centre of a circle of radius 5 cm. Its length is

   1. 4 cm
   2. 6 cm
   3. 8 cm
   4. 10 cm
   Show answer

   Answer: (c) 8 cm. 2√(25 − 9) = 2 × 4 = 8 cm.

5. An arc subtends 100° at the centre. The angle it subtends at a point on the remaining circle is

   1. 80°
   2. 100°
   3. 200°
   4. 50°
   Show answer

   Answer: (d) 50°. It is half the central angle.

6. In a cyclic quadrilateral ABCD, ∠A = 70°. Then ∠C is

   1. 70°
   2. 90°
   3. 110°
   4. 20°
   Show answer

   Answer: (c) 110°. Opposite angles add up to 180°.

7. Two chords AB and CD are equal. Which statement must be true

   1. they are parallel
   2. they are diameters
   3. they intersect at the centre
   4. they are equidistant from the centre
   Show answer

   Answer: (d) they are equidistant from the centre. Equal chords are at equal distances from the centre.

8. An exterior angle of a cyclic quadrilateral is 105°. The interior opposite angle is

   1. 105°
   2. 75°
   3. 180°
   4. 52.5°
   Show answer

   Answer: (a) 105°. The exterior angle equals the interior opposite angle.