# NCERT Class 9 Mathematics Chapter 5 I’m Up and Down, and Round and Round: Summary, Concepts, and Revision Notes | SwaVid Chapter 5 develops the geometry of circles: how a circle is defined, how its chords and arcs behave, and how those facts explain circumcircles and cycli... Canonical: https://swavid.com/maths/class/9/chapter/i-m-up-and-down-and-round-and-round Source: https://swavid.com/maths/class/9/chapter/i-m-up-and-down-and-round-and-round # I’m Up and Down, and Round and Round ## Concepts in this chapter ## The ideas this chapter keeps returning to ## Where this chapter usually trips people up ## Find the gaps before this chapter finds them for you. ## Related chapters ### Circle definitions and basic elements Chapter 5 · Class 9 Mathematics Chapter 5 develops the geometry of circles: how a circle is defined, how its chords and arcs behave, and how those facts explain circumcircles and cyclic quadrilaterals. Your study route Key topics A circle is the set of points in a plane at the same distance from a fixed centre. That distance is the radius; a chord joins two points on the circle and a diameter is a chord through the centre. Example Section 5.1 defines a circle, radius, chord, and diameter. Watch out A diameter is a special chord, not a separate kind of line segment, and its length is twice the radius. A circle is the set of points in a plane at the same distance from a fixed centre. That distance is the radius; a chord joins two points on the circle and a diameter is a chord through the centre. Tap the card for an example Example Section 5.1 defines a circle, radius, chord, and diameter. Why it matters These names and the locus definition are the vocabulary used in every later theorem in the chapter. Watch out A diameter is a special chord, not a separate kind of line segment, and its length is twice the radius. Ask at home Ask the learner to mark the centre, one radius, one chord, and one diameter on a drawn circle. A circle is the locus of points at a fixed distance from a centre. The chapter builds from centre, radius, chord, and diameter to reflection and rotational symmetry, circles through two or three points, and the circumcircle of a triangle. It then proves relationships between equal chords, central angles, perpendicular distances from the centre, and angles subtended by arcs. The final ideas are concyclicity and the fact that opposite angles of a cyclic quadrilateral add to 180 degrees. Chapter summary A circle is the locus of points at a fixed distance from a centre. The chapter builds from centre, radius, chord, and diameter to reflection and rotational symmetry, circles through two or three points, and the circumcircle of a triangle. It then proves relationships between equal chords, central angles, perpendicular distances from the centre, and angles subtended by arcs. The final ideas are concyclicity and the fact that opposite angles of a cyclic quadrilateral add to 180 degrees. What you should learn Keep these close A circle is the locus of points at a fixed distance from a fixed centre. A radius joins the centre to the circle; a chord joins two points on the circle; a diameter is a chord through the centre. Every diameter is a line of reflection symmetry, and a circle has rotational symmetry through any angle. Infinitely many circles pass through two fixed points; their centres lie on the perpendicular bisector of the joining segment. Three non-collinear points determine a unique circumcircle. Equal chords subtend equal angles at the centre, and the converse holds. The perpendicular from the centre to a chord bisects the chord, and equal chords are equidistant from the centre. The longer of two unequal chords is closer to the centre. The angle subtended by an arc at the centre is twice the angle it subtends at a point on the remaining circle. An angle subtended by a diameter at a point on the circle is 90 degrees. A quadrilateral is cyclic when its vertices lie on one circle; opposite angles of a cyclic quadrilateral sum to 180 degrees. Common confusions It is easy to think Every chord is a diameter. The clearer idea A diameter is only the chord that passes through the centre. Other chords do not pass through the centre and are shorter than the diameter. It is easy to think The centre angle and the angle at the circle standing on the same arc are equal. The clearer idea The angle at the centre is twice the angle subtended by that arc at a point on the remaining circle. It is easy to think A line from the centre to a chord always bisects the chord. The clearer idea It bisects the chord when it is perpendicular to the chord; equivalently, the perpendicular from the centre bisects the chord. It is easy to think Any four points determine a cyclic quadrilateral. The clearer idea Four points are concyclic only when they lie on one circle. Equal angles on the same side of a segment or supplementary opposite angles can provide the required evidence. Before you push ahead Most stuck chapters trace back to one earlier idea. Check the prerequisites first, or let SwaVid adapt this chapter to the way your child learns. Keep exploring Class 9 Source - Define a circle as the locus of points at a fixed distance from a centre. - Identify radius, chord, diameter, arc, centre, circumcircle, and circumcentre. - Explain the symmetry of a circle and construct circles through two or three points. - Use the relationships between equal chords, central angles, and distances from the centre. - Use the angle-in-a-semicircle and angle-subtended-by-an-arc results. - Recognise concyclic points and solve cyclic-quadrilateral angle problems. - Define a circle as the locus of points at a fixed distance from a centre. - Identify radius, chord, diameter, arc, centre, circumcircle, and circumcentre. - Explain the symmetry of a circle and construct circles through two or three points. - Use the relationships between equal chords, central angles, and distances from the centre. - Use the angle-in-a-semicircle and angle-subtended-by-an-arc results. - Recognise concyclic points and solve cyclic-quadrilateral angle problems. - NCERT Class 9 Mathematics textbook: Ganita Manjari : Chapter 5 sections 5.1–5.8 and chapter summary ## Key Links - [NCERT Mathematics](https://swavid.com/maths) - [Class 9](https://swavid.com/maths/class/9) - [Start with concepts](https://swavid.com/maths/class/9/chapter/i-m-up-and-down-and-round-and-round) - [Check a prerequisite](https://swavid.com/learning-debt-identifier) - [Concepts](https://swavid.com/maths/class/9/chapter/i-m-up-and-down-and-round-and-round) - [Practice](https://swavid.com/maths/class/9/chapter/i-m-up-and-down-and-round-and-round/practice-questions) - [NCERT Solutions](https://swavid.com/maths/class/9/chapter/i-m-up-and-down-and-round-and-round/ncert-solutions) - [Find the gaps before this chapter](https://swavid.com/learning-debt-identifier) - [Learn this chapter adapted to you](https://swavid.com/learning-style-test) - [Previous chapter 4 . 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