Cartesian Coordinate System is used to describe the position of a point in a plane using
coordinates in coordinate geometry. Coordinates are the way of telling the position of a point using numbers.

Rene Discartes invented
the Coordinate System and the term Cartesian Coordinate System is named after the name of Discartes
mathematician to honour the inventor.

Cartesian geometry system introduces **coordinate axes**, **coordinates** and **quadrants** to describe
the position of a point in a plane and its distances from other points.

Let’s learn about these new terms next.

Cartesian coordinate system uses a plane which has two mutually perpendicular lines which are called as
**coordinate axes**. Out of the two perpendicular lines, one line is in horizontal position and another line
is in
vertical position. The figure below shows how the two coordinate axes look like.

Note

Mind the words **axis** and **axes**, how they are used. Axis is used to represent one axis, it is
singular. On the other hand, axes
is used to represent more than one axis, which is plural.

Horizontal axis in coordinate geometry

Vertical axis in coordinate geometry

The horizontal line marked with XX^{‘} is called as **x-axis** and vertical line marked as
YY^{‘} is
called as **y-axis**. Both axes, x-axis and y-axis intersect each other at point O and are perpendicular to
each
other. The point O is called as **origin**.

So, x-axis has two sides, one side lies on right side of y-axis and can be read as OX. The second side lies on
left side of y-axis and is read as OX^{‘}. Similarly, y-axis has two sides top and bottom of x-axis. The
top side of y-axis is read as OY and bottom side of y-axis is OY’.

The OX on x-axis and OY on y-axis called as
positive directions of x-axis and y-axis respectively. Similarly, the OX^{‘} on x-axis and
OY^{‘} on y-axis called as negative directions of x-axis and y-axis respectively.

Directions of axis in coordinate geometry

Scaling of axes is meant by marking the x-axis and y-axis with numbers which are placed at equal distances. The
ray OX on x-axis has positive numbers (1, 2, 3, 4, ….) and ray OX^{‘} has negative numbers (-1, -2,
-3, -4,
….) only. Similarly,

The ray OY on y-axis has positive numbers (1, 2, 3, 4, ….) and ray OY^{‘} has negative numbers (-1,
-2, -3,
-4, ….) on it.
In other words, we can say that positive numbers lie in the directions of ray OX and ray OY and the negative
numbers lie
in the direction of ray OX^{‘} and ray OY^{‘}.

The two coordinate axes x-axis and y-axis in coordinate geometry divide the plane into four parts. These four
parts are called as **quadrants**.

Quadrants in coordinate geometry

In above figure, x-axis and y-axis is dividing the plane into four parts which are XOY, X^{‘}OY,
X^{‘}OY^{‘} and XOY^{‘}.

XOY is called as **first quadrant**.

X^{‘}OY is called as **second quadrant**.

X^{‘}OY^{‘} is called as **third quadrant**.

XOY^{‘} is called as fourth **quadrant**.

Also, the four quadrants are numbered as I, II, III and IV anticlockwise starting from first quadrant XOY and
the last quadrant as X^{‘}OY^{‘}.

Any point in a plane can lie in any of the four quadrants I, II, III or IV. The position of a point is in the
plane is called as **coordinates** of that point. Coordinates of the point in a quadrant is determined by
knowing
its perpendicular distances from its nearest x-axis and y-axis.

Coordinates of a point is written by writing its perpendicular distance in brackets in the format of (x,y),
where x is the perpendicular distance of the point from x-axis and y is the perpendicular distance of the point
from y-axis.

The x-axis coordinate of a point is called as **abscissa**.

The y-axis coordinate of a point is called as **ordinate**.

Coordinates of a point can be written only in specific order i.e (x,y). First we write abscissa followed by
ordinate and seperated by a comma.

Note

Abscissa and ordinate of a point cannot be interchanged.

i.e. (x,y) ≠ (y,x)

Moreover, (x,y) = (y,x) if x = y

Let’s take an example to understand it precisely

Coordinates of a point in coordinate geometry

Example

From the above figure, we can see

P is the point that lie in first quadrant and is marked has P.

The perpendicular distance PN of point P from y-axis is 3 units, measured along the positive direction
of x-axis as OM which is 3 units.

The perpendicular distance PM of point P from x-axis is 4 units, measured along the positive direction
of y-axis as ON which is 4 units.

So, the perpendicular distance of point P from y-axis is 3 and x-axis is 4, so the coordinate of the point P
can be written as (3,4).

Note

Coordinates of origin are always written as (0,0) because perpendicular distance of origin from x-axis is zero and from y-axis is also zero.

Note

The ordinate of any point on x-axis is 0 i.e coordinate of any point on x-axis is (x,0).

The abscissa of any point on y-axis is 0 i.e coordinate of any point on y-axis is (0,y).

As we have read above, the x-axis XOX’ and y-axis YOY’ divide the coordinate plane into four quadrants.

The ray OX is regarded as positive x-axis.

The ray OY is regarded as positive y-axis.

The ray OX^{‘} is regarded as negative x-axis.

The ray OY^{‘} is regarded as negative y-axis.

So, any point that lies in first quadrant will always have positive abscissa and positive ordinate.

Any point that lie in second quadrant has negative abscissa and positive ordinate.

Any point that lie in third quadrant has negative abscissa and negative ordinate.

Any point that lie in fourth quadrant has positive abscissa and negative ordinate.

Please refer to the following figure, how the signs of abscissa and ordinate looks like in the four
quadrants.

Sign conventions in quadrants in coordinate geometry

So, In the figure:

first quadrant has points with sign (+,+).

second quadrant has points with sign (-,+).

third quadrant has points with sign (-,-).

fourth quadrant has points with sign (+,-).

The branch of mathematics in which coordinate system is used to solve geometric problems is known as coordinate geometry.

Horizontal line is called as x-axis.

Vertical line is called as y-axis.

The x coordinate of a point is called as abscissa.

The y coordinate of a point is called as ordinate.

**Solution**

Point P(3,4) lies in the first quadrant. So we move 3 units along OX and 4 units in upward direction
i.e along OY.

Point Q(-5,3) lies in second quadrant. So we move 5 units along OX' and 3 units in upward direction
i.e along OY.

Point R(-5,0) lies on negative direction of x-axis at the distance of 5 units from origin.

Point S(6,0) lies on positive direction of x-axis at the distance of 6 units from origin.

**Solution**

Point A(3,-4) lies in fourth quadrant. So we move 3 units along OX and 4 units in downward direction
i.e along OY'.

Point B(-3,-5) lies in third quadrant. So we move 3 units along OX' and 5 units in downward direction
i.e along OY'.

Point C(0,5) lies on positive direction of y-axis at the distance of 5 units from origin.

Point D(0,-2) lies on negative direction of y-axis at the distance of 2 units from origin.

**Solution**

The figure joined by these points is a square.

AB = BC = CD = DA = 6 units

side of square = 6 units

Perimeter of square= 4 x 6 = 24 sq. units

**Solution**

AB = CD = 7 units

BC = AD = 4 units

So, ABCD is a rectangle

Area of rectangle = length x breadth

= 7 x 4 = 28 sq. units

**Solution**

The figure is right angled triangle when we join these points.

OM = 3 units

MN = 4 units

∠OMN = 90^{0}

Area of \(\triangle OMN\) = \(\frac{1}{2}\) x base x height

= \(\frac{1}{2}\) x OM x MN

= \(\frac{1}{2}\) x 3 x 4

= 6 sq. units

- abscissa
- ordinate
- origin
- coordinates

- Thales
- Euclid
- Pythagoras
- Rene descartes

- x-axis and y-axis
- y-axis and x-axis
- abscissa and ordinate
- ordinate and abscissa

- True
- False
- Maybe
- None of these

- First quadrant
- Second quadrant
- Third quadrant
- Fourth quadrant

- x = -4 and y = 4
- x = -4 and y = -4
- x = 3 and y = 6
- x = -1 and y = 6