Distance Between Two Points

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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 what is the distance formula.


Distance Between Any Two Points

(xA, yA) and (xB, yB)

Distance

(1, 2) and (3, 4)

2.8284

(1, 3) and (-2, 9)

6.7082

(1, 2) and (5, 5)

5

(1, 2) and (7, 6)

7.2111

(1, 1) and (7, -7)

10

(13, 2) and (7, 10)

10


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. Coordinates of a point is 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 that the distance of the point from the origin, and on the y-axis is of the form (0, a), where a is that 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 the 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 : \[AB = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}\]


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 = √(16+9) = √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:

\[AB = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}}\]

FAQ (Frequently Asked Questions)

Question 1) How do You Find the Distance Between 2 Points?

Answer) 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.

Question 2) What is the Distance Between 4 and 17?

Answer) 21 units

The linear distance between – 4 and 17 on the given number line is 21 units.

Question 3) Is the Point on the Line?

Answer) To determine that if a point is on the line we can simply change the x and y coordinates into the equation. Another way to unravel the matter would be to graph the given line and see if it falls on the line. Plugging in will give which may be a truth, so it's on the given line.

Question 4) How do You Calculate the Y - Intercept?

Answer) Using the "slope-intercept" form of the line's equation (y = mx + b), we solve for b (which is the y-intercept you're looking for). So replace the known slope for the variable m, and change the known points of the coordinates for x and y, respectively, in the slope-intercept equation. That will let us find b.