Coordinate Geometry (or Analytical Geometry) is a study of geometric figures by plotting them in the coordinate axes. It was invented by French mathematician René Descartes when he tried to describe the path of a fly crawling along criss-cross beams on the ceiling while he lay on his bed. Thus, Cartesian coordinates are named after him which revolutionized mathematics by providing the first systematic link between geometry and algebra.
Let’s understand what coordinate geometry and Cartesian system are with examples.
What is Coordinate Geometry?
Coordinate geometry (or analytic geometry) is defined as the study of geometry using coordinate points. Using coordinate geometry, it is possible to find the distance between two points, divide lines in the $m:n$ ratio, find the mid-point of a line, calculate the area of a triangle in the Cartesian plane, etc.
What is Cartesian Plane?
A Cartesian plane is defined as a plane formed by the intersection of two coordinate axes that are perpendicular to each other. The horizontal axis is called the $x$-axis and the vertical one is the $y$-axis. These axes intersect with each other at the origin whose location is given as $(0, 0)$. Any point on the cartesian plane is represented in the form of $(x, y)$. Here, $x$ is the distance of the point from the $y$-axis and $y$ is the distance from the $x$-axis.
A Cartesian plane divides the plane space into two dimensions and is useful to easily locate the points. The two axes of the coordinate plane are the horizontal $x$-axis and the vertical $y$-axis. These coordinate axes divide the Cartesian plane into four quadrants, and the point of intersection of these axes is the origin $(0, 0)$.
Any point in the coordinate plane is referred to by a point $(x, y)$, where the $x$-value is the position of the point with reference to the $x$-axis, and the $y$-value is the position of the point with reference to the $y$-axis.
The properties of the point represented in the four quadrants of the coordinate plane are:
- The origin $\text{O}$ is the point of intersection of the $x$-axis and the $y$-axis and has the coordinates $(0, 0)$.
- The $x$-axis to the right of the origin $\text{O}$ is the positive $x$-axis and to the left of the origin, $\text{O}$ is the negative $x$-axis.
- The $y$-axis above the origin $\text{O}$ is the positive $y$-axis, and below the origin, $\text{O}$ is the negative $y$-axis.
- The point represented in the $\text{I}^{st}$ quadrant $(x, y)$ has both positive values and is plotted with reference to the positive $x$-axis and the positive $y$-axis.
- The point represented in the $\text{II}^{nd}$ quadrant is $(-x, y)$ is plotted with reference to the negative $x$-axis and positive $y$-axis.
- The point represented in the $\text{III}^{rd}$ quadrant $(-x, -y)$ is plotted with reference to the negative $x$-axis and negative $y$-axis.
- The point represented in the $\text{IV}^{th}$ quadrant $(x, -y)$ is plotted with reference to the positive $x$-axis and negative $y$-axis.
Cartesian Coordinates
In the Cartesian system, a point is represented as $\text{P}(x, y)$, where $x$ is called $x$-coordinate and $y$ as $y$-coordinate.
The $x$-coordinate of a point is its perpendicular distance from the $y$-axis. It is measured along the $x$-axis which is positive along the positive direction and negative along the negative direction. In the above figure, point $\text{P}$, it is $+4$ on the positive $x$-axis. This $x$-coordinate is called the abscissa.
The $y$-coordinate of a point is its perpendicular distance from the $x$-axis. It is measured along the $y$-axis. In the above figure point $\text{P}$, it is $+3$ on the positive $y$-axis. This $y$-coordinate is called the ordinate.
Examples on Cartesian Coordinates
Example 1: What is the distance of point $\text{P}(5, 3)$ from the $x$-axis and $y$-axis?
The $y$-coordinate of $\text{P}(5, 3)$ is $3$ therefore distance of point $\text{P}$ from $x$-axis is $3$ units and the $x$-coordinate of $\text{P}(5, 3)$ is $5$ therefore distance of point $\text{P}$ from $y$-axis is $5$ units.
Example 2: Name the quadrants in which the following points lie.
a. $(-5, 6)$
b. $(2, 3)$
c. $(4, -7)$
d. $(-1, -9)$
e. $(0, 4)$
f . $(-5, 0)$
a. The $x$-coordinate of the point $(-5, 6)$ is negative and the $y$-coordinate is positive and is of the form $(-, +)$, therefore the point $(-5, 6)$ lies in the $\text{II}^{nd}$ quadrant.
b. The $x$-coordinate of the point $(2, 3)$ is positive and the $y$-coordinate is also positive and is of the form $(+, +)$, therefore the point $(2, 3)$ lies in the $\text{I}^{st}$ quadrant.
c. The $x$-coordinate of the point $(4, -7)$ is positive and the $y$-coordinate is negative and is of the form $(+, -)$, therefore the point $(4, -7)$ lies in the $\text{IV}^{th}$ quadrant.
d. The $x$-coordinate of the point $(-1, -9)$ is negative and the $y$-coordinate is also negative and is of the form $(-, -)$, therefore the point $(-1, -9)$ lies in the $\text{III}^{rd}$ quadrant.
e. The $x$-coordinate of the point $(0, 4)$ is zero and is of the form $(0, a)$, therefore the point $(0, 4)$ lies on the $y$-axis and since $y$-coordinate is positive the point lies on positive $y$-axis.
f. The $y$-coordinate of the point $(-5, 0)$ is zero and is of the form $(a, 0)$, therefore the point $(-5, 0)$ lies on the $x$-axis and since $x$-coordinate is negative the point lies on negative $x$-axis.
Representing Geometrical Figures on a Cartesian Plane
A line segment is represented in a Cartesian plane by showing its endpoints as coordinates of a point.
In the above figure the endpoints of a line segment $\overline{\text{PQ}}$ are $\text{P}(x_1, y_1)$ and $\text{Q}(x_2, y_2)$.
A rectilinear geometrical figure is made up of a number of line segments generally known as an edge or a side of a figure. The point of intersection of any two sides is a vertex of a figure which is represented by a Cartesian coordinate.
The above figure represents a $\triangle \text{ABC}$, with coordinates of vertices as $\text{A}(x_1, y_1)$, $\text{B}(x_2, y_2)$, and $\text{C}(x_3, y_3)$.
Similarly the above figure represents a quadrilateral $\text{ABCD}$, with coordinates of vertices as $\text{A}(x_1, y_1)$, $\text{B}(x_2, y_2)$, $\text{C}(x_3, y_3)$, and $\text{D}(x_4, y_4)$.
Mirror Image of Points
The mirror image of a point is obtained when a point is reflected on either $x$-axis, $y$-axis, or origin. For a point $\text{P}(x, y)$,
a. the mirror image when reflected on $x$-axis is $\text{P}^{‘}(x, -y)$
b. the mirror image when reflected on $y$-axis is $\text{P}^{‘}(-x, y)$
c. the mirror image when reflected on origin is $\text{P}^{‘}(-x, -y)$
The figure below shows a $\triangle \text{ABC}$ reflected on $y$-axis to get an image $\triangle \text{A’B’C’}$.
Practice Problems
- What is Coordinate Geometry?
- Is Analytical Geometry the same as Coordinate Geometry?
- What is a Cartesian plane?
- How a point is represented in a Cartesian plane?
FAQs
What is meant by coordinate geometry?
Coordinate geometry also known as analytical geometry is defined as the study of geometry using coordinate points.
What is the use of coordinate geometry?
Coordinate geometry is used to find the distance between two points, dividing lines in a given ratio, finding the mid-point of a line, calculating the area of a triangle in the Cartesian plane, etc.
Who is the father of coordinate geometry?
Coordinate geometry that uses Cartesian coordinates are named after René Descartes. He is also regarded as the father of coordinate geometry.
How is coordinate geometry used in real life?
There are numerous applications of coordinate geometry in real life. The maps we use to locate places: google maps, and physical maps, are all based on the coordinate system. Further, it is helpful in large-scale land projects to draw the land maps to scale. The Naval engineers use coordinate systems, to locate any point in the seas.
Conclusion
Coordinate geometry also known as analytical geometry is defined as the study of geometry using coordinate points. It has wide applications in creating and reading maps, and navigation.
