how to find distance between two points

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Think of the distance between any two points as a line. The length of this line can be found by using the distance formula: ${\sqrt {(}}(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2})$ .

Steps

1
Take the coordinates of two points you want to find the distance between. Call one point Point 1 (x1,y1) and make the other Point 2 (x2,y2). It does not terribly matter which point is which, as long as you keep the labels (1 and 2) consistent throughout the problem.^[1]
- x1 is the horizontal coordinate (along the x axis) of Point 1, and x2 is the horizontal coordinate of Point 2. y1 is the vertical coordinate (along the y axis) of Point 1, and y2 is the vertical coordinate of Point 2.
- For an example, take the points (3,2) and (7,8). If (3,2) is (x1,y1), then (7,8) is (x2,y2).
2

Know the distance formula. This formula finds the length of a line that stretches between two points: Point 1 and Point 2. The linear distance is the square root of the square of the horizontal distance plus the square of the vertical distance between two points.^[2] More simply put, it is the square root of: $(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}$

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3
Find the horizontal and vertical distance between the points. First, subtract y2 - y1 to find the vertical distance. Then, subtract x2 - x1 to find the horizontal distance. Don't worry if the subtraction yields negative numbers. The next step is to square these values, and squaring always results in a positive number.^[3]
- Find the distance along the y-axis. For the example points (3,2) and (7,8), in which (3,2) is Point 1 and (7,8) is Point 2: (y2 - y1) = 8 - 2 = 6. This means that there are six units of distance on the y-axis between these two points.
- Find the distance along the x-axis. For the same example points (3,2) and (7,8): (x2 - x1) = 7 - 3 = 4. This means that there are four units of distance separating the two points on the x-axis.
4

Square both values. This means that you will square the x-axis distance (x2 - x1), and that you will separately square the y-axis distance (y2 - y1).
5

Add the squared values together. This will give you the square of the diagonal, linear distance between your two points. In the example of the points (3,2) and (7,8), the square of (8 - 2) is 36, and the square of (7 - 3) is 16. 36 + 16 = 52.
6
Take the square root of the equation. This is the final step in the equation. The linear distance between the two points is the square root of the sum of the squared values of the x-axis distance and the y-axis distance.^[4]
- To carry on the example: the distance between (3,2) and (7,8) is sqrt (52), or approximately 7.21 units.
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Question

How do I find the horizontal distance between (3, 4) and (8, 4)?

Subtract 3 from 8 since both are at 4 on the y axis. So distance is: 8-3=5.
Question

What is the distance from the x-axis to (7,-2)?

This is an ambiguous question. I will assume you mean the shortest distance. Then, your second point will be (7,0) because the line that goes through (7,0) and (7,-2) is perpendicular to the x-axis. So your answer is 2.
Question

What is the distance between (2, 3) and (-8,12)?

Using the distance formula shown in the above article, find the horizontal distance between the two points by subtracting (-8) from 2, which is 10. Then find the vertical distance between the points by subtracting 12 from 3, which is -9. We then add together the squares of those two distances: 3² + (-9)² = 9 + 81 = 90. Find the square root of that sum: √90 = 9.49. That's the distance (in "units") between the two points.
Question

Where will I need this besides my test?

It's not likely you will use this technique in a real-life application. It is a way to practice using graphs and the Pythagorean theorem.
Question

When both points have negative X and Y positions, how do I fill in the formula?

You still fill in the formula the same way, remembering that the negative signs are part of the formula. The negative numbers squared become positive, so there should not be any problem in the end.
Question

What is the midpoint of 45, 972 and 66, 191?

The x-coordinate of the midpoint is half the distance between 45 and 66: 66 - 45 = 21. Half of 21 is 10½. Add 10½ to 45 to get the midpoint's x-coordinate, 55½. The y-coordinate of the midpoint is half the distance between 972 and 191: 972 - 191 = 781. Half of 781 is 390½. Add 390½ to 191 to get the midpoint's y-coordinate, 581½. So the midpoint is (55½, 581½).
Question

In finding the distance between two points (horizontally or vertically), is the formula used either Xsub1 -Xsub2, or Xsub2 - Xsub1?

Because all you care about is distance, not direction, you can subtract in either order. You just want to know how far apart the two points are, and subtracting in either direction will tell you. That's true of both the horizontal and vertical directions.
Question

What is the distance between (4,6) and (-5,8)?

Let (x_1, y_1) = (4,6) and (x_2, y_2) = (-5,8). The distance formula is sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). Simply plug in to the formula, and you obtain sqrt(85).
Question

What is the distance between (2, -4) and (-5, 3)?

The distance formula is sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). 3-(-4)=7-5-2=-7 (7)^2=49 (-7)^2=49 sqrt(49+49)=9.8.
Question

If the distance between two point is 7 and the points are 5,2 and x,4, how do I find the value of x?

The distance formula is sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). 7= sqrt((x-5)^2 + (4-2)^2)....> Square both side....>49=(x-5)^2 + (4-2)^2 .=>49=(x-5)^2+4=>49-4=(x-5)^2 =>45=(x-5)^2=>sqrt(45)=(x-5)=>x=6.7-5=>x=1.7

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It doesn't matter if you get a negative number after subtracting y2 - y1 or x2 - x1. Because the difference is then squared, you will always get a positive distance in your answer.^[5]

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Article SummaryX

To find the distance between two points on a line, take the coordinates of the two points. Label one as Point 1, with the coordinates x1 and y1, and label the other Point 2, with the coordinates x2 and y2. Plug these values into the distance formula, which is the square of X2 minus X1 plus the square of Y2 minus Y1, then the square root of that result. To see the distance formula written out, read on!

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