What happens to the Distance Formula if the x-coordinates are equal?
See solution.
Practice makes perfect
Let's try our friend's claim with an example.
An Example
We can draw a line segment between A(3,5) and B(5,2). They have the same x-coordinates but different y-coordinates.
Now, we can count the number of units between A and B to determine the length of the segment. It's 3.
Do we get the same length if we subtract the y-coordinates?
5-2=3
Yes! Your friend seems to be correct. However, he does not say in which order we should subtract. If we change the order of the y-coordinates, we no longer get the same answer.
2-5=-3
We cannot have a negative distance. -3 is not the length of AB.
Comparing this to the Distance Formula
Let's take a look at the general form of the Distance Formula.
AB=sqrt((x_2-x_1)^2+(y_2-y_1)^2)
If the x-coordinates are equal, the first term will always subtract to be 0.
sqrt((0)^2+(y_2-y_1)^2) ⇒ sqrt((y_2-y_1)^2)
This is only equal to y_2-y_1 if y_2-y_1 is positive.
Conclusion
Our friend is, therefore, partly correct. We can find the length of a segment between two points with the same x-coordinates by subtracting the y-coordinates, but only if we subtract the lesser y-coordinate from the greater one.
Or, the order does not matter if we take the absolute value of the subtraction. Looking at the example from above, we can see that this is true.
AB=|5-2| & ⇒ AB=| 3|=3
AB=|2-5| & ⇒ AB=|-3|=3
