These Class 9 Maths Notes and Chapter 5 I’m Up and Down and Round and Round Class 9 Ganita Manjari Notes are designed according to the latest CBSE syllabus.
Class 9 Maths Chapter 5 I’m Up and Down and Round and Round Notes
Class 9 Maths Ganita Manjari Chapter 5 Notes
Class 9 I’m Up and Down and Round and Round Notes
→ Introduction: Humanity has always been fascinated by the shapes of the things around them. There are instances in the cave paintings where the Sun is depicted using a circle. We can see numerous geometric patterns which include triangles, squares, circles and ovals, in the cave paintings of Gudahandi in Odisha.
Nature is full of such shapes such as we see the concentric layers of wood that form annually as a tree grows as their growth rings.
Circles form when raindrops fall on water. The cross-section of a plant stem and the inflorescence of a sunflower are also circular in shape. The full moon and the sun also look circular.
→ A Circle, It’s Centre and Radius:
A circle is the set of all points on the plane that are equidistant from a given point on that plane. The fixed point is called its centre and the distance of any point on the circle from this fixed point is its radius.
In the above figure 1, 0 is the centre, A, X, B, Y are the points on the circle. The distance of any of these points will be the radius of the circle.
So, OA,OX,OB and OY are its radii (the plural of radius).
Note: The set of points that satisfy a given condition is also called the locus of points that satisfy the given conditions. So, we can define a circle as ‘the locus of all those points which are at equal distances form a fixed point’.
→ Chord and Diameter of A Circle:
In figure 1, AC is a chord of the circle and XZ is its diameter. So, a chord can be defined as ‘the line segment joining any two points on the circle’. Diameter is the largest chord of a circle. Its length is twice that of the radius of the circle. Diameter of a circle always passes through its centre. Diameter XZ passes through the centre O, in figure.
→ Symmetries of a Circle:
Circle is a perfectly symmetrical shape. A rotating wheel looks the same at all times. There are many rotational symmetries for a circle. All diameters are line of reflection symmetry. A circle has rotational symmetry about its centre, through any angle. As there can be many diameters in a circle, so there are many lines of reflection symmetry in a circle.
→ Number of Circles:
Many numbers of circles can pass through a given point or through a pair of given points. Infinitely many circles can pass through two given points and centres of these circles lie on the perpendicular bisectors of the line segment joining these two points. A unique circle passes through three non collinear points.
Note: Any three or more points are said to be collinear points if they lie on a line. Obviously, no circle can be drawn through three collinear points.
→ Circumcircle and Its Circumcentre:
For a triangle, if a circle passes through its vertices, then the circle is called Circumcircle. Its centre is called the Circumcentre and lies at the intersection of the perpendicular bisectors of the line segments joining the points. Conversely, we say that the triangle is inscribed in a circle. For an acute- angled triangle, the circumcentre lies inside the triangle, for an obtuse-angled triangle, the circumcentre lies outside the triangle and for a right-angled triangle, the circumcentre lies at the mid-point of the hypotenuse.
→ Chords and The Angles They Subtend:
Equal chords of a circle subtend equal angles at the centre of the circle.
In figure, if chords AB = DE then ∠ACB = ∠DCE.
Conversely, chords of a circle that subtend equal angles at the centre are equal. In figure, if ∠ACB = ∠DCE then AB = DE.
→ The line joining the centre of a circle and the midpoint of a chord of the circle is perpendicular to the chord.
In figure, since, AM = BM, so, CM ⊥AB,
Conversely, the perpendicular from the centre of a circle to a chord of the circle bisects the chord.
→ Distance of Chords from the Centre:
Chords of a circle having the same length are all at the same distance from the centre of the circle.
In figure, since AB = GF, hence, CE = CH, i.e., AB and GF are at equal distances from the centre.
Conversely, if CE = CH, then AB = GF, i.e., chords of a circle that are equidistant from the centre have equal length.
→ Larger chord of a circle is nearer to the centre of the circle.
In figure, AB > DE => CF < CG.
Note: The chord nearest to the centre is the chord containing the centre. Its distance from the centre of the circle is zero, so the diameter is the greatest chord of any circle.
→ Arc of a Circle:
An arc of a circle is a connected portion of the circle. It is defined by two points on the circle, called the end points of the arc, and the curve connecting them along the circle’s edge.
In figure, AYB and AXB are arcs. Arc AXB is less than the semicircle (half of the circle) so, it is minor arc. On the other hand, we can define a major arc as the arc which is longer than the semicircle. AYB is the major arc.
→ Angle Subtended by an Arc:
We define the angle subtended by the arc AB at the centre to be the measure of the angle AOB, as we sweep along the arc – so we move from OA to OB along the arc and measure the angle swept.
In figure, ∠AOB is angle subtended by the minor arc AXB and the reflex ∠AOB is the angle subtended by major arc AYB.
→ The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc at any point on the circle outside the arc.
In figure, ∠AOB = 2 × ∠AQB; ∠AOB = 2 × ∠APB; ∠AOB = 2 × ∠ARB.
→ Concyclicity of Points:
If a line segment AB joining two points A, B subtends equal angles at two other points C, D that lie on the same side of AB, then the four points lie on a circle. These four points are called concyclic points. If we join these points in that order, we get a quadrilateral called cyclic quadrilateral. Cyclic quadrilateral has all four vertices that lie on a circle.
The sum of two opposite angles of a cyclic quadrilateral is 180°. In fig. 9, ∠A + ∠C = 180° and ∠ABC + ∠ADC = 180°.
Conversely, if there exists a quadrilateral in which the sum of two opposite angles is 180°, then the quadrilateral is cyclic i.e., its four vertices lie on a circle.