Summary
Chapter 5 of NCERT Class 9 Maths, "I'm Up and Down, and Round and Round", studies circles: their definition as the locus of points equidistant from a centre, their symmetries, properties of chords and the angles they subtend, arcs, and concyclic points.
This chapter defines a circle as the set of all points in a plane equidistant from a fixed centre, with that distance being the radius. It introduces chords, diameters, and the circle's rotational and reflection symmetry. Students learn how many circles pass through one, two, or three points, leading to the unique circumcircle of a triangle and its circumcentre. The chapter proves theorems on equal chords and equal angles, the line from the centre to a chord's midpoint, and distances of chords from the centre. It closes with angles subtended by arcs, the angle in a semicircle, concyclic points, and cyclic quadrilaterals.
Key points & formulas
- 01A circle is the locus of points equidistant from a fixed centre
- 02Circles have full rotational symmetry and reflection symmetry across any diameter
- 03Infinitely many circles pass through two points; centres lie on the perpendicular bisector
- 04A unique circumcircle passes through three non-collinear points
- 05Equal chords subtend equal angles at the centre, and conversely
- 06The line from centre to a chord's midpoint is perpendicular to the chord
- 07Angle subtended by an arc at the centre is twice that at any point on the circle
- 08Opposite angles of a cyclic quadrilateral add up to 180 degrees
Frequently asked questions
01What is a circle according to NCERT Class 9 Maths Chapter 5?
A circle is the set of all points on a plane that are equidistant from a given point, called the centre. This set is also called the locus of points equidistant from the centre, and the equal distance from the centre to any point on the circle is the radius.
02What is the circumcircle and circumcentre of a triangle?
There is a unique circle passing through three non-collinear points, which form a triangle. This circle is the circumcircle, and its centre O, found where the perpendicular bisectors of the sides intersect, is called the circumcentre. It lies inside an acute triangle, outside an obtuse triangle, and at the midpoint of the hypotenuse for a right-angled triangle.
03What does the arc-angle theorem say in this chapter?
The angle subtended by an arc at the centre of the circle is double the angle subtended by the same arc at any point on the circle outside the arc. A corollary is that the angle subtended by a diameter at any point on the circle is 90 degrees.
04What is a cyclic quadrilateral and what property do its angles have?
When the four vertices of a quadrilateral are concyclic (lie on the same circle), the quadrilateral is called cyclic. The sum of each pair of opposite angles of a cyclic quadrilateral is 180 degrees, and conversely, if two opposite angles of a quadrilateral add up to 180 degrees, it is cyclic.
More chapters in Ganita Manjari
This is the complete Ganita Manjari Chapter 5 as published by NCERT — every diagram, solved example, and exercise included, free. Browse all NCERT Class 9 textbooks.
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