geometry

Foundations of Data Science: Coordinate Geometry

We are learning the basics of Data Science. This is the second article on Coordinate Geometry which is the second step of the data science foundation. Read the first article on Linear Algebra basics here.

Introduction to Coordinate Geometry

The study of geometry with coordinate points is known as Coordinate Geometry. It describes the link between geometry and algebra through graphs to solve geometric problems. Coordinate geometry is also known as Cartesian Geometry.

In coordinate geometry, the position of points on the plane is described using an ordered pair of numbers called coordinates. To locate the position of a point or to find the coordinates of a point on a plane, we require a pair of coordinate axes (the x-axis and y-axis). The distance of a point from the y-axis is called its x-coordinate, or abscissa. A point’s x-coordinate, also known as its abscissa, is its distance from the y-axis. Y-coordinate, also known as ordinate, is a point’s distance from the x-axis. Points on the x- and y-axes have coordinates of the forms (x, 0) and (0, y), respectively.

In coordinate planes, we have two axes labeled as the x-axis (horizontal axis) and y-axis (vertical axis), which are perpendicular to each other. The points at which these axes intersect are called the origin (0, 0). Coordinates are two variables (x, y) that indicate a point’s location on a plane.

Basics of Coordinate Geometry

There are four quadrants along with their respective values in the graph of coordinate planes, as shown in Figure 1.2

  • Quadrant 1 : (+x, +y)

In quadrant 1, the values of x and y are both positive.

  • Quadrant 2 : (-x, +y)

In quadrant 2, the value of x is negative and y is positive.

  • Quadrant 3 : (-x, -y) positive values of x and y

In quadrant 3, the values of x and y are both negative.

  • Quadrant 4 : (+x, -y) positive values of x and y

In quadrant 4, the value of x is positive and y is negative.

coordinate

Figure 1.2: Coordinate plane

Figure 1.2 shows the coordinate plane with a point (3,2), where ‘3’ is the x-coordinate and ‘2’ is the y-coordinate of the point on the coordinate plane.

Distance Formula

We can easily calculate the distance between two points whose coordinates are given.

Let’s first calculate the distance of a point A (x1, y1) from the origin (0, 0)

distance

Figure 1.3: Distance of a point from the origin

Let’s say the distance between point A and the origin is ‘d’. If we draw perpendiculars from point A to the x-axis and y-axis, we will get the distance from A to the x-axis as y1 and from A to the y-axis as x1. 

Now, this makes a triangle, so using the Pythagoras Theorem, we can say:

Pythagoras theorem

 

 

The above theorem can be written as follows: one point is (x1, y1), and the other is at the origin.

Pythagoras theorem

 

 

 

Thus, we can calculate the distance between any two points A (x1, y1) and B (x2, y2) in a coordinate plane using the graph in Figure 1.4

Distance between two points

 

 

 

 

 

 

 

 

 

 

 

Figure 1.4: Distance between two points

Using Pythagoras theorem in the above graph, the distance between points A and B comes out to be:

Distance between two points

 

 

The above formula is called the distance formula.

Stay Tuned!!

Keep learning and keep implementing!!

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