Cartesian Coordinate Plane

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On a graph paper draw two mutually perpendicular straight lines X'OX and YOY', intersecting each other at the point O. These lines are known as coordinate axes. The line X'OX is called the x-axis and YOY' is called the y-axis. The point O is called the origin.

The plane of the paper containing both the axes is called the Cartesian Coordinate Plane.

Coordinate Plane

Cartesian Coordinate Plane

On right hand side of the x-axis, every end point of a square represents a positive integer, as shown in the figure.

On left hand side of the x-axis, every end point of a square represents a negative integer.

Above the y-axis, every end point of a square represents a positive integer.

Below the y-axis, every end point of a square represents a negative integer.

Ordered pair:In a Cartesian coordinate plane, a pair of two numbers a and b, is called an ordered pair.

Note that (a, b) `ne` (b, a). Thus (4, 3)`ne` (3, 4).

The Cartesian Coordinate Systemuses the ordered pair a,b to specify each point on the coordinate plane.
The Cartesian Coordinate Plane: Coordinates of ...
... a Point

On a graph paper, let X'OX and YOY' be the coordinate axes.

Let P be a point on the graph paper such that P is at a distance of a units from the y-axis and b units from the x-axis. Then we say coordinates of P are P(a, b).

Here a is called the x-coordinate or abscissa of P and b is called the y- coordinate or ordinate of P.

Coordinate Plane

Quadrants : The Cartesian coordinate plane is divided by the axes into four regions called the quadrants.

Quadrants
Region Quadrant Sign of coordinates
XOY I (+, +)
YOX' II (-, +)
X'OY' III (-, -)
Y'OX IV (+, -)

The Cartesian Coordinate Plane: Plotting Points

Ex 1: Plot each of the following points on a graph paper:

i) A(5, 2) ii)B(-2, 4) iii) C(-1, -1) iv) D(4, -3)

Sol :

i) On x-axis, take 5 units to the right of y-axis and then on the y-axis, take 2 units above the x-axis. Thus we get the point A(5, 2).

ii) i) On x-axis, take 2 units to the left of y-axis and then on the y-axis, take 4 units above the x-axis. Thus we get the point A(-2, 4).

iii)i) On x-axis, take 1 unit to the left of y-axis and then on the y-axis, take 1 unit below the x-axis. Thus we get the point A(-1, -1).

iv) i) On x-axis, take 4 units to the right of y-axis and then on the y-axis, take 3 units below the x-axis. Thus we get the point A(4, -3).

Introduction about two dimensional planes:

The shape of an object located in some space is the part of that space occupied by an object, as determined by the boundary. All flat shapes in geometry are called as plan figures. Some plan figures are two dimensional. the following shapes are comes under plane geometric figures.

Triangle
Rectangle
Rhombus
kite
trapezoid etc

In this tutorial we shall see about properties of that plane figures.
Two Dimensional Planes:
1. Triangle:

equilateral triangle

It is a two dimensional plane geometry figure It has tree sides. The total sum of all three internal angles triangle is 180 degree.

Area of equilateral triangle (A) = a2 x /4

Perimeter of triangle (p) =sum of three sides of triangle.

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Example problem:

Find the area and perimeter of triangle whose side 9 cm

Solution:

Given:

Side (a) = 9 cm.

Area of the triangle (A) = a2 x `sqrt(3)` /4

= 92 x (1.732/4)

= 81 x 0.433

= 35.07 cm2

Perimeter of equilateral triangle (P) = 3a units

= 3 x 9

= 27 cm
2. Square:

square

It is a two dimensional plane geometry figure. It has four side all are equal in length. It has four right angles. The sum of internal angles is 360 degree.

Area of square (A) =a2 square units

Perimeter of the square (P) = 4 x a units

Example problem:

1. Find the area and perimeter of the square, whose side length is 23.6 meter.

Solution:

Given:

Side (a) =23.6 m

Area of square = a2 square units

= 23.6 x 23.6

Area of square = 556.96 m2

Perimeter of the square (P) = 4 x A units

Perimeter of the square = 4 x a units

= 4 x 23.6

Perimeter of the square = 94.4 meters
Two Dimensional Planes:

3. Rhombus:

rhombus

It is a two dimensional plane geometry figure .It has two pairs of parallel sides. The opposite sides of rhombus are equal in length and opposite angles are equal in measure.

Area of the rhombus (A) = b x h square units.

Perimeter of rhombus (P) = 4b units

Example problem:

Find the area of rhombus whose base 13 cm and height 7 cm.

Solution:

Given:

Base= 13 cm; height= 7 cm

Formula:

Area of the rhombus (A) = b x h square units

= 13 x 7

= 91 cm2

Area of the rhombus (A) = 91 cm2

Perimeter of rhombus (P) = 4 x base

= 4 x 13

= 52 cm

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