`color{red} ♦` Introduction

You have already studied how to locate a point on a number line.

There are many other situations, in which to find a point we are required to describe its position with reference to more than one line.

For example, consider the following situations :

`I.` In Fig. `3.1`, there is a main road running in the East-West direction and streets with numbering from West to East. Also, on each street, house numbers are marked.

To look for a friend’s house here, is it enough to know only one reference point? For instance, if we only know that she lives on Street `2`, will we be able to find her house easily?

Not as easily as when we know two pieces of information about it, namely, the number of the street on which it is situated, and the house number.

If we want to reach the house which is situated in the `2`nd street and has the number `5`, first of all we would identify the 2nd street and then the house numbered 5 on it.

In Fig. `3.1`, H shows the location of the house. Similarly, P shows the location of the house corresponding to Street number `7` and House number `4`.

II. Suppose you put a dot on a sheet of paper [Fig.3.2 (a)]. If we ask you to tell us the position of the dot on the paper, how will you do this?

Perhaps you will try in some such manner: “The dot is in the upper half of the paper”, or “It is near the left edge of the paper”, or “It is very near the left hand upper corner of the sheet”.

Do any of these statements fix the position of the dot precisely? No! But, if you say “ The dot is nearly 5 cm away from the left edge of the paper”, it helps to give some idea but still does not fix the position of the dot.

A little thought might enable you to say that the dot is also at a distance of 9 cm above the bottom line. We now know exactly where the dot is!

For this purpose, we fixed the position of the dot by specifying its distances from two fixed lines, the left edge of the paper and the bottom line of the paper [Fig.3.2 (b)]. In other words, we need two independent informations for finding the position of the dot.

Now, perform the following classroom activity known as ‘Seating Plan’.

`color{blue}text(Activity 1 (Seating Plan))` : Draw a plan of the seating in your classroom, pushing all the desks together.

Represent each desk by a square. In each square, write the name of the student occupying the desk, which the square represents.

Position of each student in the classroom is described precisely by using two independent informations:

(i) the column in which she or he sits,

(ii) the row in which she or he sits.

If you are sitting on the desk lying in the 5th column and 3rd row (represented by the shaded square in Fig. 3.3), your position could be written as (5, 3), first writing the column number, and then the row number. Is this the same as (3, 5)?

Write down the names and positions of other students in your class. For example, if Sonia is sitting in the 4th column and 1st row, write S(4,1).

The teacher’s desk is not part of your seating plan. We are treating the teacher just as an observer.

In the discussion above, you observe that position of any object lying in a plane can be represented with the help of two perpendicular lines.

In case of ‘dot’, we require distance of the dot from bottom line as well as from left edge of the paper. In case of seating plan, we require the number of the column and that of the row.

This simple idea has far reaching consequences, and has given rise to a very important branch of Mathematics known as Coordinate Geometry. In this chapter, we aim to introduce some basic concepts of coordinate geometry.

You will study more about these in your higher classes. This study was initially developed by the French philosopher and mathematician René Déscartes.

All the other streets of the city run parallel to these roads and are `200 m` apart. There are `5` streets in each direction. Using `1 cm = 200 m`, draw a model of the city on your notebook. Represent the roads/streets by single lines.

There are many cross- streets in your model. A particular cross-street is made by two streets, one running in the North - South direction and another in the East - West direction.

Each cross street is referred to in the following manner : If the 2nd street running in the North - South direction and 5th in the East - West direction meet at some crossing, then we will call this cross-street (2, 5). Using this convention, find:

(i) how many cross - streets can be referred to as (4, 3).

(ii) how many cross - streets can be referred to as (3, 4).

There are many other situations, in which to find a point we are required to describe its position with reference to more than one line.

For example, consider the following situations :

`I.` In Fig. `3.1`, there is a main road running in the East-West direction and streets with numbering from West to East. Also, on each street, house numbers are marked.

To look for a friend’s house here, is it enough to know only one reference point? For instance, if we only know that she lives on Street `2`, will we be able to find her house easily?

Not as easily as when we know two pieces of information about it, namely, the number of the street on which it is situated, and the house number.

If we want to reach the house which is situated in the `2`nd street and has the number `5`, first of all we would identify the 2nd street and then the house numbered 5 on it.

In Fig. `3.1`, H shows the location of the house. Similarly, P shows the location of the house corresponding to Street number `7` and House number `4`.

II. Suppose you put a dot on a sheet of paper [Fig.3.2 (a)]. If we ask you to tell us the position of the dot on the paper, how will you do this?

Perhaps you will try in some such manner: “The dot is in the upper half of the paper”, or “It is near the left edge of the paper”, or “It is very near the left hand upper corner of the sheet”.

Do any of these statements fix the position of the dot precisely? No! But, if you say “ The dot is nearly 5 cm away from the left edge of the paper”, it helps to give some idea but still does not fix the position of the dot.

A little thought might enable you to say that the dot is also at a distance of 9 cm above the bottom line. We now know exactly where the dot is!

For this purpose, we fixed the position of the dot by specifying its distances from two fixed lines, the left edge of the paper and the bottom line of the paper [Fig.3.2 (b)]. In other words, we need two independent informations for finding the position of the dot.

Now, perform the following classroom activity known as ‘Seating Plan’.

`color{blue}text(Activity 1 (Seating Plan))` : Draw a plan of the seating in your classroom, pushing all the desks together.

Represent each desk by a square. In each square, write the name of the student occupying the desk, which the square represents.

Position of each student in the classroom is described precisely by using two independent informations:

(i) the column in which she or he sits,

(ii) the row in which she or he sits.

If you are sitting on the desk lying in the 5th column and 3rd row (represented by the shaded square in Fig. 3.3), your position could be written as (5, 3), first writing the column number, and then the row number. Is this the same as (3, 5)?

Write down the names and positions of other students in your class. For example, if Sonia is sitting in the 4th column and 1st row, write S(4,1).

The teacher’s desk is not part of your seating plan. We are treating the teacher just as an observer.

In the discussion above, you observe that position of any object lying in a plane can be represented with the help of two perpendicular lines.

In case of ‘dot’, we require distance of the dot from bottom line as well as from left edge of the paper. In case of seating plan, we require the number of the column and that of the row.

This simple idea has far reaching consequences, and has given rise to a very important branch of Mathematics known as Coordinate Geometry. In this chapter, we aim to introduce some basic concepts of coordinate geometry.

You will study more about these in your higher classes. This study was initially developed by the French philosopher and mathematician René Déscartes.

All the other streets of the city run parallel to these roads and are `200 m` apart. There are `5` streets in each direction. Using `1 cm = 200 m`, draw a model of the city on your notebook. Represent the roads/streets by single lines.

There are many cross- streets in your model. A particular cross-street is made by two streets, one running in the North - South direction and another in the East - West direction.

Each cross street is referred to in the following manner : If the 2nd street running in the North - South direction and 5th in the East - West direction meet at some crossing, then we will call this cross-street (2, 5). Using this convention, find:

(i) how many cross - streets can be referred to as (4, 3).

(ii) how many cross - streets can be referred to as (3, 4).