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Coordinates and ratio

Coordinates and ratio

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Summary

​​Coordinates and ratio

In a nutshell

Knowing the coordinates of the end-points of a line segment is enough to describe locations on the line segment. The midpoint is just halfway between the xx- and yy-coordinates, but other positions can be found using ratios.​



Midpoints

To find the point in the middle of two other points, find the middle between the points' xx-coordinates and the middle between the points' yy-coordinates. These middle values are the xx​- and yy- coordinates of the midpoint.


Finding the middle between two numbers can be done by halving the difference between them and then adding this to the smaller of the two numbers. Or more formally, using the following procedure.


procedure

1.
Designate one of your points as point AA​ and the other as point BB​. Let point AA have the coordinates (xA,yA)(x_A,y_A)​ and point BB have the coordinates (xB,yB)(x_B,y_B).​
2.
Find the middle xx-value using the formula:
xA+xBxA2x_A+\frac{x_B-x_A}{2}​​
3.
Find the middle yy-value using the similar formula:
yA+yByA2y_A+\frac{y_B-y_A}{2}​​
4.
You now have that the midpoint has coordinates (xA+xBxA2,yA+yByA2)(x_A+\frac{x_B-x_A}{2},y_A+\frac{y_B-y_A}{2}).​



Using a ratio to locate a point 

The same idea is used here. Instead of halving the difference between the points' coordinates, you apply the given ratio.


procedure

1.
Designate one of your points as point AA​ and the other as point BB​. Let point AA have the coordinates (xA,yA)(x_A,y_A)​ and point BB have the coordinates (xB,yB)(x_B,y_B).​
2.
Suppose the new point splits the line from A to B into the ratio α:β\alpha:\beta. This means the new point is αα+β\frac{\alpha}{\alpha+\beta} of the line from A.​
3.
Find the new point's xx-coordinate using the formula:
xA+αα+β(xBxA)x_A+\frac{\alpha}{\alpha+\beta}(x_B-x_A)​​
4.
Find the new point's yy-coordinate using the similar formula:
yA+αα+β(yByA)y_A+\frac{\alpha}{\alpha+\beta}(y_B-y_A)​​
5.
You now have that the new point has coordinates (xA+αα+β(xBxA),yA+αα+β(yByA))(x_A+\frac{\alpha}{\alpha+\beta}(x_B-x_A),y_A+\frac{\alpha}{\alpha+\beta}(y_B-y_A)).​


Example 1

Take the line segment with the points A (2,8)(2,-8) and B (6,24)(-6,24) at either end. The point C is located on this line segment such that it divides the line in the ratio 1:31:3. What are the coordinates of point C?


The ratio says that point C is a quarter of the distance of the line segment from point A. This is found by dividing the first part of the ratio, 11 by the sum of both parts of the ratio, 1+31+3:

11+3=14\frac{1}{1+3}=\frac14


You want to therefore find the coordinates that are a quarter of the xx- and yy-differences from point A.


The difference between the xx coordinates is 

62=8-6-2=-8​​


To apply the ratio, multiply this by the quarter, then add it to the 22:

8×14=2-8\times\frac14=-2


2+(2)=02+(-2)=0


Thus the xx-coordinate of point C is 00. Now do the same for the yy-coordinate: the difference is

24(8)=3224-(-8)=32


Multiplying this by the quarter (from the ratio) then adding to 8-8 gives

32×14=832\times\frac14=8


8+8=0-8+8=0


So the yy-coordinate of point C is also 00. Hence point C has coordinates (0,0)\underline{(0,0)}​.​​

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FAQs - Frequently Asked Questions

How does splitting a line segment into a ratio differ from finding the midpoint?

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How do you find the midpoint of two points?

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