Maharashtra Board Class 9 Maths Part II Chapter 7 Co ordinate Geometry PDF Download

Co-ordinate Geometry

Chintu and his friends were playing cricket on the ground in front of a big building, when a visitor arrived.

Visitor: Hey Chintu, Dattabhau lives here, doesn't he?

Chintu: Yes, on the second floor. See that window? That's his flat.

Visitor: But there are five windows on the second floor. It could be any of them!

Chintu: His window is the third one from the left, on the second floor.

Axes, Origin, Quadrants

We could give the location of Dattabhau's house using two ordinal numbers. Similarly, the location of a point can be fully described using its distances from two mutually perpendicular lines.

To locate a point in a plane, a horizontal number line is drawn in the plane. This number line is called the X-axis.

Chintu's description of the location of Dattabhau's flat is in fact based on the most basic concept in Co-ordinate Geometry.

It did not suffice to give only the floor number to locate the house. Its serial number from the left or from the right also needed to be given. That is two numbers had to be given in a specific sequence. Two ordinal numbers namely, second from the ground and third from the left had to be used.

Teacher's Note

In India, we use coordinates to find addresses. Just like Chintu found Dattabhau's flat by saying "second floor, third window from left," we use coordinates (x, y) to find any point on a plane.

Exam Trick

Remember: Like a house address needs floor number and window number, a point needs two coordinates (x, y). The order matters—(2,3) is different from (3,2).

Points to Remember

Two perpendicular number lines are needed to locate a point.
The horizontal line is called the X-axis.
The vertical line is called the Y-axis.
The point where the axes meet is called the origin (O).
Every point has two coordinates written as (x, y).

Let's Learn

Axes, Origin, Quadrants

Another number line intersecting the X-axis at point marked O and perpendicular to the X-axis, is the Y-axis. Generally, the number O is represented by the same point on both the number lines. This point is called the origin and is shown by the letter O.

On the X-axis, positive numbers are shown on the right of O and negative numbers on the left.

On the Y-axis, positive numbers are shown above O and negative numbers below it.

The X and Y axes divide the plane into four parts, each of which is called a Quadrant. As shown in the figure, the quadrants are numbered in the anti-clockwise direction. The points on the axes are not included in the quadrants.

Rene Descartes (1596-1650)

Rene Descartes, a French mathematician of the 17th Century, proposed the co-ordinate system to describe the position of a point in a plane accurately. It is called the Cartesian co-ordinate system. Obviously the word Cartesian is derived from his name. He brought about a revolution in the field of mathematics by establishing the relationship between Algebra and Geometry.

The Cartesian co-ordinate system is the foundation of Analytical Geometry. La Geometric was Descartes' first book on mathematics. In it, he used algebra for the study of geometry and proposed that a point in a plane can be represented by an ordered pair of real numbers. This ordered pair is the 'Cartesian Co-ordinates' of a point.

Co-ordinate geometry has used in a variety of fields such as Physics, Engineering, Nautical Science, Seismology and Art. It plays an important role in the development of technology in Geogebra. We see the inter-relationship between Algebra and Geometry quite clearly in the software Geogebra, the very name being a combination of the words 'Geometry' and 'Algebra'.

Teacher's Note

Rene Descartes invented coordinates to solve real problems. Today, GPS in phones uses coordinates to show your location—just like the Cartesian system Descartes created.

Exam Trick

Remember: Cartesian = Descartes. The word comes from his name. In the coordinate system, we always write x-coordinate first, then y-coordinate, like (x, y).

Points to Remember

Rene Descartes created the Cartesian coordinate system.
He connected Algebra with Geometry.
Coordinates use an ordered pair (x, y).
The X-axis is horizontal and the Y-axis is vertical.
Quadrants are numbered counter-clockwise starting from the top right.

The Co-ordinates of a Point in a Plane

The point P is shown in the plane determined by the X-axis and the Y-axis. Its position can be determined by its distance from the two axes. To find these distances, we draw seg PM ⊥ X-axis and seg PN ⊥ Y-axis.

Co-ordinate of point M on X-axis is 2 and co-ordinate of point N on Y-axis is 3.

Therefore x co-ordinate of point P is 2 and y co-ordinate of point P is 3.

The convention for describing the position of a point is to mention x co-ordinate first. According to this convention the order of co-ordinates of point P is decided as 2, 3. The position of the point P in brief, is described by the pair (2, 3).

The order of the numbers in the pair (2, 3) is important. Such a pair of numbers is called an ordered pair.

To describe the position of point Q, we draw seg QS ⊥ X-axis and seg QR ⊥ Y-axis. The co-ordinate of point Q on the X-axis is 3 and the co-ordinate on the Y-axis is 2. Hence the co-ordinates of point Q are (3, 2).

Ex. Write the co-ordinates of points E, F, G, T in the figure alongside.

Solution:

The co-ordinates of point E are (2, 1)

The co-ordinates of point F are (3, 3)

The co-ordinates of point G are (4, 2)

The co-ordinates of point T are (3, 1)

Teacher's Note

To find coordinates of a point, count steps right or left from the origin for x, then count steps up or down for y. This is like a treasure map in a classroom.

Exam Trick

Remember: (x, y) = (across, up). Always write x-coordinate first. Count horizontal distance first, then vertical distance.

Points to Remember

Every point has exactly one pair of coordinates.
The order of coordinates matters: (2,3) is not the same as (3,2).
The first number is the x-coordinate (horizontal distance).
The second number is the y-coordinate (vertical distance).
We write coordinates as an ordered pair in brackets.

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