NCERT Class 9 Ganita Manjari Chapter 05 Im Up and Down and Round and Round PDF Download

Chapter 5: I'm Up and Down, and Round and Round

Humanity has always been fascinated by the shapes of the things around them. In some early cave paintings, the sun is depicted as a circle. In the cave paintings of Gudahandi in Odisha, one sees numerous geometric patterns including triangles, squares, circles, and ovals. These shapes were likely inspired by what humans saw in nature. Can you recognise the origin of the shapes in Fig. 5.1?

[Figure 5.1: Raindrops on water, cross-section of a plant stem, and inflorescence of a sunflower, See in your textbook]

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 sun also look circular.

[Figure 5.2: Moon and Sun (during a total solar eclipse), See in your textbook]

What properties are common to all circles, big and small? You have studied one such property in Grade 7. Humans must have noticed it after observing many circular patterns in nature. Every circle has a centre. All points on the circle are at equal distance from the centre. We turn this observation around and make it the definition of a circle.

Activity

List some objects from nature that resemble a circle.

Think and Reflect

Jamuna has a circular piece of paper. She is trying to locate its centre. Amina gives her a suggestion. She follows the instructions and is thrilled to find that it works. Can you guess what Amina told her?

5.1 Definitions

When we talk of mathematical shapes such as circles, triangles and squares, we always assume that the figures are drawn on a piece of paper - a two-dimensional plane.

[Figure 5.3: Circle, Centre A, Chord BC, See in your textbook]

A circle is the set of all points on the plane that are equidistant from a given point on that plane. The set of points that satisfy a given condition is also called the locus of points that satisfy the condition. Using this term, a circle can also be described as the locus of points that are equidistant from a given point. The given point is the centre of the circle. The distance from the centre to any point on the circle is the radius of the circle.

In Fig 5.3, A is the centre of the circle. All points on the plane at a distance equal to the length of AB from A form a circle with centre A and radius equal to the length of AB.

Let B and C be two points on the circle. The line segment BC is called a chord of the circle. The angle subtended by the chord BC at the centre is angle BAC. A chord passing through the centre of a circle is called a diameter.

Teacher's Note

Remember that the radius is a distance (a number), not a line segment. When you write "the radius is 5 cm", you mean the distance from the centre to any point on the circle is 5 cm. The diameter is always twice the radius: if the radius is 5 cm, the diameter is 10 cm.

5.2 Symmetries of a Circle

What makes circles so appealing is that they are perfectly symmetrical. Say you are looking at a wheel of a vehicle. You see a point of the wheel touching the ground. When you look at the wheel again after some time, you again see a point of the wheel touching the ground. Can you tell if the two points are the same point? There is no way you can tell; a rotating wheel looks the same at all times! We say the circle has complete rotational symmetry: rotate it by any angle and it looks exactly the same.

Draw a circle on the paper and cut along the circle. Fold the circular paper so that the boundaries overlap, then open it. You see a crease; it is a line of reflection symmetry of the circle. Does this line pass through the centre of the circle? It does. It is a diameter of the circle. All diameters are lines of reflection symmetry.

Think and Reflect

1. What are the rotational symmetries of a square? How many lines of reflection symmetry does it have? What about a regular pentagon? A regular hexagon?
2. What is the length of the longest chord in a circle of radius 5 units? Is there a smallest chord?
3. The locus of points at a given distance from a given point is a circle. What can we say about the locus of points equidistant from two given points?
   Hint: We know that any point that is equidistant from two given points A and B lies on the perpendicular bisector of AB. Does this make the perpendicular bisector the locus? For this, we have to show that all the points on the perpendicular bisector are equidistant from A and B.

Teacher's Note

When you fold a circle to find a line of reflection symmetry, any diameter you create divides the circle into two identical halves. This is why every diameter of a circle is a line of reflection symmetry. In an exam, if asked to identify all lines of symmetry of a circle, remember that there are infinitely many (all possible diameters), not just a few like in a square.

5.3 How Many Circles?

Now that we have defined a circle and listed some of its properties, let us ask this question: Given two points A and B on a plane, how many circles pass through A and B?

If a circle passes through A and B, it has a centre, say O. The lengths OA and OB are equal. Is there another point with this property? Yes, the midpoint of segment AB. With the midpoint as centre, a circle passing through A and B can be drawn. Its radius is half the length of AB, and AB is a diameter.

Key Points

• A circle is the set of all points that are at the same distance (the radius) from a fixed point (the centre).
• The diameter is the longest chord in a circle and passes through the centre; it is always twice the radius.
• A circle has perfect rotational symmetry: it looks the same when rotated by any angle. Every diameter of a circle is a line of reflection symmetry.

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