Coordinate Geometry- class 9

Introduction

• The study of geometric figures using coordinate axes is known as coordinate geometry.

• Figures like circles, ellipses, polygons, and straight lines can be drawn and scaled on the coordinate axes with ease.

• Additionally, coordinate geometry facilitates algebraic operations and the study of the properties of geometric figures.

• Every location on this globe has coordinates that make it simple for us to find them on a map.

• Our planet's coordinate system is made up of fictitious lines known as latitudes and longitudes.

• We have the coordinate axes with the horizontal x-axis and the vertical y- axis, which are similar to placing the point on a piece of paper or a plane.

Coordinate Geometry

• In order to present geometric forms in a two-dimensional plane and learn about their properties, coordinate geometry is a crucial area of mathematics.

• In order to get a basic concept of coordinate geometry, we'll try to learn about the coordinate plane and a point's coordinates here.

• The coordinate planes consist of two planes. One is a horizontal plane, i.e., the x-axis and the other is a vertical plane, i.e., the y-axis and these two planes divide the plane into four quadrants.

• The origin (0,0) is the point where these two planes intersect.

• Any location in this plane is represented as (x, y) where x and y represent the horizontal and vertical position, respectively, from the origin (0,0).

Abscissa and Ordinates

A coordinate is an address that aids in locating a site in space. These coordinates are represented as (x, y) in a 2D plane or a coordinate is divided into abscissa and ordinates.

• Abscissa − In the coordinate (x, y), the x value is called the Abscissa of the coordinate, which is the horizontal (x-axis) distance from the origin

• Ordinates − In the coordinate (x, y), the y value is called the Ordinates of the coordinate, which is the vertical (y-axis) distance from the origin.

Axes

• A line that is used to take or mark measurements is known as an axis in mathematics.

• The two crucial axes of the coordinate plane are the x and y axes.

• A horizontal number line is the x-axis, while a vertical number line is the y- axis.

• The coordinate plane is created by the perpendicular intersection of these two axes.

• The ordinate and abscissa are other names for the x-axis and y-axis, respectively.

Quadrants

In a coordinate plane, we have four quadrants because the x-axis and y-axis divide the plane into four parts, which are called quadrants. Below, the figure shows the four quadrants marked as I,II, III, and IV and their coordinate signs.

Allocation of Coordinates

In order to allocate the points in the plane, we need coordinates. The x- coordinate represents the distance of the point from the y-axis whereas the y- coordinate represents the distance of the point from the x-axis.

For example − If we are looking to allocate the point (2,4) on the coordinate plane, then we will go 2 units in the positive x direction and from there we will go 4 units in the positive y direction, hence we have reached our wanted location i.e., (2,4).

Distance from Axes

The distance from axes simply means the distance of the coordinate point (x,y) from the axis i.e., The x- coordinate represents the distance of the point from the y-axis whereas the y-coordinate represents the distance of the point from the x- axis.

Distance Formula

To calculate the distance between two given points, let's say (a,b) and (p, q), you have to find the square root of the difference between the x coordinates and the y coordinates of the two given points.

$$\mathrm{D\:=\:\sqrt{(p\:-\:a)^{2}\:+\:(q\:-\:b)^{2}}}$$

Solved Examples

Example 1 − Which quadrant or axis does each of the coordinates (-2, 4), (-3, -1), (-1, 0), (1-2), and (-3, -5) belong to? To check the correctness of your answer, locate them on the Cartesian plane.

Solution −

The following are the signs of the quadrant

Quadrant I: (+, +)

Quadrant II: (-, +)

Quadrant III: (-, -)

Quadrant IV: (+, -)

We will match the given coordinates with the above quadrant signs.

So,

(-2, 4): Quadrant II since it follows the rule (-, +)

(-3, -1), (-3, -5): Quadrant III since it follows the rule (-, -)

(1, -2) − Quadrant IV since it follows the rule (+, -)

(-1, 0) Does not lie in any Quadrant rather on the negative x-axis.

For more clarity, refer to the below graph.

Example 2 − Find the distance between the points (-2,4) and (4,2) using the distance formula.

Solution − To calculate the distance between the given two points, we will use the distance formula. It goes as follows, the distance D between two points $\mathrm{A\:(x_{1}\:,\:y_{1})}$ and $\mathrm{B\:(x_{2}\:,\:y_{2})}$ is given by the square root of the sum of squares of the differencesnof the respective parts of the coordinates of the two points.

$$\mathrm{D\:=\:\sqrt{(x_{2}\:-\:x_{1})^{2}\:+\:(y_{2}\:-\:y_{1})^{2}}}$$

Substituting the respective values of the two points we have,

$$\mathrm{D\:=\:\sqrt{(4\:-\:(-2))^{2}\:+\:(2\:-\:4)^{2}}}$$

Simplifying

$$\mathrm{D\:=\sqrt{36\:+\:4}\:=\:\sqrt{40}\:units}$$

The distance between the given two points is √40 units.

Conclusion

• The coordinate planes consist of two planes. One is a horizontal plane, i.e., the x-axis and the other is a vertical plane, i.e., the y-axis.

• Any location in this plane is represented as (x, y) where x and y represent the horizontal and vertical position, respectively, from the origin (0, 0).

• In the coordinate (x, y), the x value is called the abscissa

• In the coordinate (x, y), the y value is called the ordinates.

• A line that is used to take or mark measurements is known as an axis in mathematics.

• In a coordinate plane, we have four quadrants because the x-axis and y-axis divide the plane into four parts, which are called quadrants.

• The distance from axes simply means the distance of the coordinate point (x, y) from the axis

FAQs

1. What are the coordinates?

Any location in the plane is represented as (x, y) where x and y represent the horizontal and vertical position, respectively, from the origin (0, 0), and this (x, y) is called coordinates.

2. How many quadrants are there in a coordinate plane?

In a coordinate plane, we have four quadrants because the x-axis and y-axis divide the plane into four parts.

3. What is Abscissa?

In the coordinate (x, y), the x value is called the abscissa of the coordinate, which is the horizontal (x-axis) distance from the origin.

4. What is the distance from axes in a plane?

The distance from axes simply means the distance of the coordinate point (x, y) from the axis i.e., The x- coordinate represents the distance of the point from the y-axis whereas the y-coordinate represents the distance of the point from the x-axis.

5. What are Ordinates?

In the coordinate (x, y), the y value is called the ordinates of the coordinate, which is the vertical (y-axis) distance from the origin.