Contents

# Coordinates|Definition & Meaning

## Definition

Coordinates are representative values of the **distance** of a point or object from a specific point or object used as a reference. The most common coordinates are the** latitude** and **longitude** coordinates which give the location of an object anywhere on the earth concerning the reference of the prime meridian located in **Greenwich**.Â

## Theoretical Explanation of CoordinatesÂ

Coordinates are to address a **point** on a plane by using a pair of numbers. They are also known as **math coordinates,** which we use to make graphs. These coordinates are written in the form of **ordered pair** consisting of an x value and a y value (x, y) to give us a point to be used in a map or give directions.

This helps us to get the precise location of a point. The following example shows a point that tells us the exact position of a plane.

Figure 1 – Representation of Coordinates

In this example, we can see the ordered pair containing both x and y coordinates, which we use to determine the quadrant of a plane. When representing coordinates on a plane, we must first understand the position of points on a cartesian plane concerning the mathematical signs, which can be a+ or a-.Â

The Coordinates divide the cartesian plane into 4 quadrants using axes XXâ€™ and YYâ€™. Following are the 4 quadrants:

- The region XOY is called the
**1st quadrant**here; both X and Y are positive. - The region Xâ€™OY is called the
**2nd quadrant**here; only X is positive and Y is negative. - The region Xâ€™OYâ€™ is called the
**3rd quadrant**here; X is negative and Y is positive. - The region Yâ€™OX is called the
**4th quadrant**here; both X and Y are negative.

Figure 2 – Division of Quadrants in a Plane

## Coordinates of a Point

A point, when plotted on a plane, uses an ordered pair consisting of values of both the **x-axis **and** y-axis** generally. To understand the concepts better, take a look at the following example:

In the following example, we can see that the following points have been plotted:

A = (2, 3)

B = (-4, 4)

C = (-2, -1)

D = (4, -2)

We can see that these points have been plotted by using both their x coordinates and y coordinates.

Figure 3 – Points plotted in a plane representing their coordinates

## How To Read a Coordinate?

To plot a point, we must first know how to deal with a coordinate, and for that, we must be aware of certain points:Â

- To find the x coordinate, look at the
**perpendicular distance**of the point from the y axis - To find the y coordinate, find the perpendicular distance of the point from the x-axisÂ
- After both of the coordinates have been found, write them in a
**round bracket**which will be separated by a comma in this form (x, y) - The first number, which represents the x-axis, is known as
**absicca** - The second number, which represents the y-axis, is called the
**ordinate**

## Uses of Coordinates

Coordinates hold significant importance in geometry and mathematics. They are used to perform different operations. Some of its uses are as below:

- The and y coordinates are used to find the
**slop of a line** - The x and y coordinates are used to find the
**distance**between two points and the midpoint of a line - The x and y coordinates help us in
**locating**the exact position on the map - The x and y coordinates are used to find the
**equation of a line**

## Types of Coordinates

If we go into the depth of coordinates, we get to know that 2 types of coordinate systems are used on a daily basis. Let us take a look at them.

**Cartesian Coordinate System**

It is a system that uses **mutually perpendicular** lines on a plane to denote the coordinates of a point. These lines may also b referred to as axes.

To locate a point, we can use these perpendicular lines, and the point where they both meet can be the desired point on the plane when we’re dealing with the x-axis and y-axis. The coordinates are referred to in the form of A(x, y), where x represents the number on the x-axis and y represents the y-axis.

**Polar Coordinate System**

It is a system where the origin, where both axes meet, holds quite important as it is taken as the reference point known as a **pole**. The pole is used to find the points depending on the distance of the point from the origin.

The coordinates are referred to in the form of **(r,Î¸),** where r represents the radius of the line and Î¸ represents the angle which is forms between the line segment and the axis.

**Coordinates on a 2D Plane**

When we are dealing with a point on a plane of x-axis and y-axis, and the intersecting point is (0, 0), then it is known as a 2D plane, and any point plotted on such a plane consisting of perpendicular distances from both x and y axes will have 2D coordinates.

We write them as **A(x, y),** where x represents the values of the x-axis and y represents the values of the y-axis.Â

**Coordinates on a 3D Plane**

When we need to find a point in a 3D plane, we use the 3D coordinate system. This system consists of 3 axes; the x-axis, y-axis, and z-axis, which all are perpendicular at a single point known as the origin. We represent it in the following form: **A(x, y, z)**

## Solved Examples of Coordinates

**Example 1**

If the centroid of a triangle has the following vertices (5, 2), (6,3), (8,2). Find the x and y coordinates.

**Solution**

The vertices of the triangle are: (5, 2), (6,3), (8,2)

Â (x_{1}, y_{1}) = (5, 2)

(x_{2}, y_{2}) = (6,3)

(x_{3}, y_{3}) = (8,2)

Coordinates = [(x_{1} + x_{2} + x_{3})/3, (y_{1} + y_{2} + y_{3})/3]

Putting values:

= [(5 + 6 + 8)/3, (2 + 3 + 2)/3]

= (19/3), (7/3)Â

= (6.3, 2.3)

**Example 2**

Harvey needs to find the x and y coordinates of the midpoint of a line with endpoints (-4, 4), (2, 8).

**Solution**

The points that we have are:

(x_{1}, y_{1}) = (-4, 4)

(x_{2}, y_{2}) = (2, 8)

To find the midpoints we will use the following formula:

MidPoint = [(x_{1} + x_{2})/2, (y_{1} + y_{2})/2]

= [(-4 + 2)/2, (4 + 8)/2]

= (-2/2, 12/2)

= (-1, 6)

*Images/mathematical drawings were created with GeoGebra.*