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Coordinates and ratio

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Explainer Video

Tutor: Toby

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 xxx- and yyy-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' xxx-coordinates and the middle between the points' yyy-coordinates. These middle values are the xxx​- and yyy- 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 AAA​ and the other as point BBB​. Let point AAA have the coordinates (xA,yA)(x_A,y_A)(xA​,yA​)​ and point BBB have the coordinates (xB,yB)(x_B,y_B)(xB​,yB​).​
2.
Find the middle xxx-value using the formula:
xA+xB−xA2x_A+\frac{x_B-x_A}{2}xA​+2xB​−xA​​​​
3.
Find the middle yyy-value using the similar formula:
yA+yB−yA2y_A+\frac{y_B-y_A}{2}yA​+2yB​−yA​​​​
4.
You now have that the midpoint has coordinates (xA+xB−xA2,yA+yB−yA2)(x_A+\frac{x_B-x_A}{2},y_A+\frac{y_B-y_A}{2})(xA​+2xB​−xA​​,yA​+2yB​−yA​​).​



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 AAA​ and the other as point BBB​. Let point AAA have the coordinates (xA,yA)(x_A,y_A)(xA​,yA​)​ and point BBB have the coordinates (xB,yB)(x_B,y_B)(xB​,yB​).​
2.
Suppose the new point splits the line from A to B into the ratio α:β\alpha:\betaα:β. This means the new point is αα+β\frac{\alpha}{\alpha+\beta}α+βα​ of the line from A.​
3.
Find the new point's xxx-coordinate using the formula:
xA+αα+β(xB−xA)x_A+\frac{\alpha}{\alpha+\beta}(x_B-x_A)xA​+α+βα​(xB​−xA​)​​
4.
Find the new point's yyy-coordinate using the similar formula:
yA+αα+β(yB−yA)y_A+\frac{\alpha}{\alpha+\beta}(y_B-y_A)yA​+α+βα​(yB​−yA​)​​
5.
You now have that the new point has coordinates (xA+αα+β(xB−xA),yA+αα+β(yB−yA))(x_A+\frac{\alpha}{\alpha+\beta}(x_B-x_A),y_A+\frac{\alpha}{\alpha+\beta}(y_B-y_A))(xA​+α+βα​(xB​−xA​),yA​+α+βα​(yB​−yA​)).​


Example 1

Take the line segment with the points A (2,−8)(2,-8)(2,−8) and B (−6,24)(-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 1:31:31:3. 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, 111 by the sum of both parts of the ratio, 1+31+31+3:

​11+3=14\frac{1}{1+3}=\frac141+31​=41​


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


The difference between the xxx coordinates is 

−6−2=−8-6-2=-8−6−2=−8​​


To apply the ratio, multiply this by the quarter, then add it to the 222:

​−8×14=−2-8\times\frac14=-2−8×41​=−2


2+(−2)=02+(-2)=02+(−2)=0


Thus the xxx-coordinate of point C is 000. Now do the same for the yyy-coordinate: the difference is

24−(−8)=3224-(-8)=3224−(−8)=32


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

32×14=832\times\frac14=832×41​=8


−8+8=0-8+8=0−8+8=0


So the yyy-coordinate of point C is also 000. Hence point C has coordinates (0,0)‾\underline{(0,0)}(0,0)​​.​​

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Coordinates in the first quadrant

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FAQs - Frequently Asked Questions

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

Instead of halving the difference between the points' coordinates, you apply the given ratio. Technically, finding the midpoint is just splitting the line into the ratio 1:1.

How do you use a ratio to find the scale factor?

If a line AB is split into the ratio α:β, then to go from A to the new point, multiply the difference between A and B by α/(α+β). ​

How do you find the midpoint of two points?

​To find the point in the middle of two other points, find the middle between the points' x-coordinates and the middle between the points' y-coordinates.

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