Remember that the distance on a number line is given by the formula: [tex]d=|x_{2}-x_{1}| [/tex]
where
[tex]d[/tex] is the distance between the points.
[tex]x_{2}[/tex] is the second point in the number line.
[tex]x_{1}[/tex] is the first point in the number line.
Now, suppose we are trying to find the distance between the points -1 and -5; our first point is -1, so [tex]x_{1}=-1[/tex], and our second point is [tex]-5[/tex], so [tex]x_{2}=-5[/tex]. Suppose we are going to find the distance between the two point without using absolute value:
[tex]d=x_{2}-x_{1}[/tex]
[tex]d=-5-(-1)[/tex]
[tex]d=-5+1[/tex]
[tex]d=-4[/tex]
Look what we have here, a negative distance! Since distances cannot be negative, we must use absolute value to always get postie distances between two point on a umber line:
[tex]d=|x_{2}-x_{1}|[/tex]
[tex]d=|-5-(-1)|[/tex]
[tex]d=|-5+1|[/tex]
[tex]d=|-4|[/tex]
[tex]d=4[/tex]
Now, the distance between two coordinates on a plane is given by the formula: [tex]d= \sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2} [/tex]
[tex]where [/tex]
[tex]d[/tex] is the distance between the two coordinates
[tex](x_{1},y_{2})[/tex] are the coordinates of the first point
[tex](x_{2},y_{2})[/tex] are the coordinates of the second point
Notice that [tex](x_{2}-x_{1})^2[/tex] and [tex](y_{2}-y_{1})^2[/tex] are squared, so it doesn't matter if we get a negative distance because a negative number raised to an even power (like 2) is always positive; therefore we don't need absolute value in this case because we won't ever get a negative distance.
