Plotting a Point in the Plane if Its Coordinates are Given: A coordinate plane is a two-dimensional plane formed by the intersection of a vertical line known as \(y-\)axis and a horizontal line known as \(x-\)axis. These are the perpendicular lines that intersect each other at a point known as the origin.

The plane is called the cartesian plane, and the lines are known as the coordinate axes. The coordinate axes divide the plane into four parts which are known as the quadrants. In this article, we will learn how to plot a point in the cartesian plane.

Coordinate Plane

A coordinate plane is a two-dimensional surface divided by two lines, intersecting each other at right angles. The horizontal line is called the \(x-\)axis, and the vertical line is called the \(y-\)axis. These two axes intersect at a point called the origin.

The numbers on a coordinate grid are used to locate points. A coordinate plane can be used to graph points, lines, and many more. It acts as a map and yields precise directions from one point to another.

What is a Point on a Coordinate Plane?

Each point in a coordinate plane is represented by an ordered pair \((x, y),\) written in parentheses, corresponding to the \(x-\)coordinate and the \(y-\)coordinate. These coordinates can be positive, zero, or negative, depending on the location of the point in the respective quadrants.

How Do You Find the Points on a Coordinate Plane?

To find the point on the coordinate plane, you have to follow the steps given below:

1. First, you have to locate the point.
2. Then, you have to find the quadrant by looking at the symbols of its \(x\) and \(y\) coordinates.
3. The \(x-\)coordinate or abscissa of the point is then determined by reading the number of units right/left of the origin along the \(x-\)axis. 
4. Finally, you have to find the \(y-\)coordinate or the point by reading the number of units above or below the origin along the \(y-\)axis of parallel to it.

Example: Now, look at the diagram given below:

1. First, observe the red dot given on the coordinate graph.
2. It is placed at the second quadrant.
3. Now, here the point is \(3\) units far from the origin along the negative \(x-\)axis.
4. And, the point is \(2\) units far from the origin parallel to the positive \(y-\)axis.

Hence, the point on the graph given has coordinates \((-3, 2).\)

Quadrant

The quadrant can be refered to as the region or the part of a cartesian plane obtained when the two axes intersect one another at right angles. It is utilised to determine the position of the point in a plane.

Four Quadrants: The \(x-\)and the \(y-\)axes divide the plane into four graph quadrants:

1. The upper right-hand corner of the plane is the first quadrant. In this quadrant, both \(x\) and \(y-\)coordinates will be positive.
2. The upper left-hand corner of the plane is the second quadrant. In this quadrant, the \(x-\)coordinate is negative, and the \(y-\)coordinate is the positive.
3. The lower left-hand corner of the plane is the third quadrant. In this quadrant, both \(x\) and \(y-\)coordinates are negative.
4. The lower right-hand corner of the plane is the fourth quadrant. In this quadrant, the \(x-\)coordinate is positive, and the \(y-\)coordinate is negative.

Quadrants on the Coordinate Plane

The quadrant can be defined as a region or the part of a cartesian or the coordinate plane obtained when the two axes are intersected perpendicularly.

1. First quadrant is \(x>0, y>0\)
2. Second quadrant is \(x<0, y>0\)
3. Third quadrant is \(x<0, y<0\)
4. Fourth quadrant is \(x>0, y<0\)

How Do You Write Coordinate Points?

Coordinates are always written in the brackets, with the two numbers split by the comma. Coordinates are the ordered pairs of the numbers; the first number shows the point on the \(x-\)axis, called the abscissa, and the second is the point on the \(y-\)axis, called the ordinate.

Plotting a point on the coordinate plane: We will see how to plot a point on the coordinate plane with an example: 

Example: We shall plot the point \(P=(5, 6.)\) To plot the point in the coordinate plane, you have to follow the given steps below:

1. You have to draw two perpendicular lines, \(x-\)axis and \(y-\)axis.
2. From the point of origin, you have to move 5 units to the right side along the positive \(x-\)axis.
3. Now, you have to move \(6\) units up, parallel to the positive \(y-\)axis
4. Here, you mark the point of intersection and mark it as \((5, 6).\)

Note that point \(P\) is in the first quadrant. Also, this is called the positive coordinate plane, as the value of both the coordinates for any point in this quadrant will be positive.

Uses of Point on a Coordinate Plane

You can use the coordinate plane 

a) To locate or plot a point and draw a line segment, triangle, quadrilateral, and other geometrical figures. Their reflection about the \(x-\)and \(y-\)axis and the origin can also be located.

b) Graphs of algebraic equations can be drawn on a coordinate plane.

c) On a larger scale, the concept of the coordinate plane can be applied to draw maps of countries, continents and the whole world. It has huge applications in geography.

Solved Examples

Q.1. Locate the points \((5,0),(0,5),(2,5),(5,2),( – 3,5),( – 3, – 5),(5, – 3)\) and \(\left( {6,1} \right)\).
Ans: Taking \({\rm{1}}\,{\rm{cm = 1}}\,{\rm{unit,}}\) we draw the \(x-\)axis and the \(y-\)axis. Dots show the positions of the points in the given diagram.

Q.2. Plot the following points in the cartesian plane (Use the scale: \(x – {\rm{axis}} = 1\;{\rm{cm }}\) and \(y – {\rm{axis}} = 1\;{\rm{cm}}\))

\(x\)	\(-3\)	\(0\)	\(-1\)	\(4\)	\(2\)
\(y\)	\(7\)	\(-3.5\)	\(-3\)	\(4\)	\(-3\)

Ans: From the above-given table, the ordered pairs formed are given as
\((-3, 7), (0, -3.5), (-1, -3), (4, 4)\) and \((2, -3)\)
The point \((-3, 7)\) lies in the \({{\rm{2}}^{{\rm{nd}}}}\) quadrant
The point \((0, -3.5)\) lies on the negative \(y-\)axis
The point \((-1, -3)\) lies in the \({{\rm{3}}^{{\rm{rd}}}}\) quadrant
Point \((4, 4)\) lies in the \({{\rm{1}}^{{\rm{st}}}}\) quadrant
The point \((2, -3)\) lies in the \({{\rm{4}}^{{\rm{th}}}}\) quadrant.

Q.3. Locate the coordinates \((3, 5)\) and \((5, -4)\) in the cartesian coordinate system.
Ans: You have to draw the coordinate axes, the \(x-\)axis and \(y-\)axis
Then, select the unit like \(1\) centimetre displays one unit on both the \(x-\)axis and \(y-\)axis.
Now, the coordinate \((3, 5)\) displays the distance from the origin to the positive \(x-\)axis is \(3\) units and the distance from the origin to the positive \(y-\)axis is \(5\) units.
Now, mark the above points on the coordinate plane, and name it as \(“P”.\)
Here, the coordinate \((5, -4)\) displays the distance from the origin to the positive \(x-\)axis is \(5\) units, and the distance from the origin to the negative \(y-\)axis is \(4\) units.
Mark the coordinate point in the plane and name it \(“Q”.\)
Hence, point \(P (3, 5)\) lies in the first quadrant, and point \(Q (5, -4)\) lies in the fourth quadrant.

Q.4. Locate the given point in the coordinate plane: \((-9, 4)\)
Ans: Given that the \(x\) and \(y\) coordinate of the point are \(-9\) and \(4,\) respectively.
Here, the given point \((-9, 4)\) is located in the coordinate plane, as shown in the diagram.

Q.5. Locate the given point in the coordinate plane: \((7, -8)\)
Ans: Given that the \(x\) and \(y\) coordinate of the point are \(7\) and \(-8,\) respectively. Hence, the point lies in the fourth quadrant.
Now, move seven units on the right side from the origin and then move eight units vertically downwards to locate the point \((7, -8).\)
The given point \((7, -8)\) is located in the coordinate plane, as shown in the diagram.

Summary 

In the given article, we have discussed coordinate planes and a point on the coordinate plane with examples. Then we have talked about how to find or locate a point on a coordinate plane. We glanced at how to write a point on a coordinate plane followed by uses of plotting a point on a coordinate plane. Finally, we have provided solved examples along with a few FAQs.

Learn the Concepts of Cartesian System

Frequently Asked Questions (FAQs)

Q.1. How is a point plotted in the plane?
Ans: Let us take the example of plotting a point \((4, 7).\)
You have to follow the given steps to plot the point in the plane:
You have to draw two perpendicular lines, \(x-\)axis and \(y-\)axis.
From the point of origin, you have to move \(4\) units to the right side along the positive \(x-\)axis.
Now, you have to move \(7\) units up, along with the positive \(y-\)axis
Here, you mark the point of intersection and mark it as \((4, 7).\)

Q.2. How do you find a point on the coordinate plane?
Ans: To find a point on the coordinate plane, you have to follow the given steps below:
First, you have to locate the point.
Then, you have to find the quadrant by looking at the symbols of its \(x\) and \(y\) coordinates.
Next, you have to find the \(x-\)coordinate or the abscissa of the point by reading the number of the units the point that is to the right/left of the origin along the \(x-\)axis.
Finally, you have to find the \(y-\)coordinate or the point by reading the number of units above or below the origin along or parallel to the \(y-\)axis.

Q.3. What is the distance between two points?
Ans: The distance between two points is the length of the line segment that connects the two given points. In coordinate geometry, the distance can be calculated by identifying the length of the line segment joining the given two coordinates.

Q.4. How can you graph a point with rational coordinates on a coordinate plane?
Ans: To graph a point \((a, b)\) with rational coordinates on the coordinate plane, you have to follow the given steps:
First, you have to locate \(a\) on the \(x-\)axis on either side whether a is positive or negative.
Then, from point \(a\) on the \(x-\)axis, if \(b\) is positive, move upwards \(b\) units, and if \(b\) is negative, then you have to move down \(b\) units.
This is how you draw the point \((a, b).\)

Q.5. How do you plot ordered pairs on a coordinate plane?
Ans: To plot ordered pairs on the coordinate plane, you have to draw a dot on the coordinates corresponding to the ordered pair.
Start from the origin, the \(x-\)coordinate lets you know how many steps you have to take to the right side (positive) or left side (negative) on the \(x-\)axis, and the \(y-\)coordinate lets you know to have steps to move upwards (positive) or downwards (negative) on the \(y-\)axis.

We hope this detailed article on the concept of plotting a point in the plane helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!