Distance Between Two Points

For any two points, there's exactly one line segment connecting them. The distance between the two points is known to be the length of the line segment connecting them. Note that the distance between the two points is always positive. Segments which have the same length are called congruent segments. We can easily calculate the distance between two points. Take the coordinates of two points you would like to seek out space between. Call one point Point 1 (x1, y1) and make the opposite Point 2 (x2, y2). It doesn't terribly matter which point is which, as long as you retain the labels (1 and 2) consistent throughout the matter. Let’s discuss the distance formula.

Distance between Any Two Points

How to Use Pythagoras Theorem to Seek Out the Distance Between Two Positions?

Consider the Following Situation.

A boy walked towards the north 30 meters and took a turn to the east and walked for 40 meters more.

How do we calculate the shortest distance between the initial place and the final place? A pictorial representation of the above situation is

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The initial point is supposedly A and therefore the final point is C. The distance between points A, B is 30 m and between points B, C is 40 m. The shortest distance between the two points A and C is AC.

This is calculated using the Pythagoras theorem as follows:

\[AC^{2} = AB^{2} + BC^{2}\]

\[AC^{2} = 30^{2} + 40^{2} = 900 + 1600\]

AC = 50 m

Hence, we got the space between the beginning point and therefore the endpoint. In the same way, the space between two points during a coordinate plane is additionally calculated using the Pythagorean theorem or right-angles triangle theorem.

Before getting to derive the formula for distance between two points during a coordinate plane, allow us to understand what are the coordinate points and the way to locate them within the Cartesian plane.

Coordinates of a Point

2D geometry deals with the coordinates of the points, distance between the points, etc. The coordinates of a point are a pair of numbers that exactly define the location of that point in the coordinate plane.

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Coordinates of the point P in the two-dimensional plane is (x, y) which means, P is x units away from the y-axis and y units away from the x-axis.

Coordinates of some extent on the x-axis are of the shape (a, 0), where a is the distance of the point from the origin, and on the y-axis is of the form (0, a), where a is the distance of the point from the origin.

What is the Distance Formula for Two Points?

Let’s discuss, what is the distance formula is used to find the distance between two points when we already know the coordinates. The points could be present alone in the x-axis or y-axis or in both axes. Let us take into account that there are two points, let’s say A and B in an XY plane. The coordinates of point A are ( x1,y1), and of B are ( x2,y2). Then the formula to seek out the space/distance between two points PQ is given by:

Distance formula : \[AB=\sqrt{(x2-x1)^{2} + (y2-y1)^{2}}\]

How to Find the Distance/Space Between the Two Points on a Coordinate Plane?

Suppose there are two points in a plane P (2, 3) and Q(-2, 0). So, how will we calculate the distance between P and Q. Thus, we will use the above equation here.

Here, x1 = 2, x2 = -2, y1  = 3 and y2 = 0.

Now, we know what is the distance formula.Now, putting up these values within the distance formula, we get;

PQ = (-2 - 2)² + (0 - 3)²

PQ = \[\sqrt{(16+9)} = \sqrt{25}\] = 5 unit.Therefore, we have found the distance between the points P and Q.

What is the Distance Between Two Points Formula in 3D Space?

If we have to find the distance between the points in a three-dimensional space, then we consider here an extra coordinate which is present in the z-axis.

Let us consider two points A(x1, y1, z1) and B (x2, y2, z2) in 3d space.

Therefore, the distance formula for these two given points is written as:

\[AB=\sqrt{(x2-x1)^{2} + (y2-y1)^{2} + (z2-z1)^{2}}\]

Geometrical Shapes

(🔴) Point

Students study various geometrical concepts right from their childhood. Geometrical concepts are introduced from the class for themselves. The first point which the students learn in geometrical concepts is point. What is a point is a dot made by the sharp point of a pencil; and it is written by a capital letter of the English alphabet.

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Uses of Points in Coordinate Geometry

The use of points in coordinate  geometry are:

It is used in fields like trigonometry, calculus, dimensional geometry, etc.

The coordinates of a point may be defined as the numbers that define the proper location on one or two-dimensional planes. There are two axes of the coordinate plane at right angles to each other. The two axes are called the x and y-axis.  The coordinates of a point tell how far along each axis the point is located. While drawing figures like a plane or solid figures points come into consideration points are used at the end of a figure to give it an exact shape. Figures like triangle square, rectangle, pentagons, prisms cubes, octagons, etc. They are useful to draw lines, line segments, ray polygons, angles, and circles.                                                                                   

Application of Coordinate Geometry in Real Life

Changing any shape or adding different colors changes the point on the coordinate plane.scanner and photocopying machine also use coordinate geometry to produce the exact image provided to it.

Coordinates of a point-Geometric applications

There are points in the Euclidean geometry that are positioned in the plane. The coordinates of the points along the x and y axis define these points. Thus, the coordinates of a point are defined as the pair of values that define the location of the point in the coordinate plane. 

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In the above diagram, the coordinates of point P in the two-dimensional plane are x and y. This means that the point P is x units far from the y axis and y units far from the x-axis. 

Coordinates of a point on the x-axis are in the form (a, 0), where a is the distance of the point from the origin, and on the y axis are in the form (a,0) where a is the distance from the point of origin.

Distance between Two Points – Using Pythagoras Theorem

Consider the following situation.

A man walked 30 meters towards the north and took a turn to the east and then walked for 40 meters. Now, calculate the shortest distance between the starting point and the final point.

The above situation can be represented as follow:

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The starting point is A and the final point is C. The distance between points A and B is 30 m and between B and C is 40 m.