Use the Distance Formula and the x-axis of the coordinate plane. Show why the distance between two points on a number line (the x-axis) is | a – b |, where a and b are the x-coordinates of the points.
How would I solve this?
so your two points are A(a,0) and B(b,0)
from A to B = √( (b-a)^2 + 0^2)
from B to A = √( (a-b)^2 + 0^2)
but (a-b)^2 = (b-a)^2 , (when you square a number the result is positive)
e.g. 5^2 = (-5)^2
so the two results are the same
thank you so much!
To solve this, we can use the distance formula:
d = | x₁ - x₂ |
In this case, the points are on a number line, which is essentially the x-axis of the coordinate plane. Therefore, our formula will be:
d = | a - b |
Now, let's see why this formula holds true:
Suppose we have two points A and B on the number line, with coordinates a and b respectively. Without loss of generality, assume a > b. We want to find the distance between these two points.
The distance between two points can be determined by finding the absolute value of the difference between their coordinates. In this case, we subtract the smaller coordinate (b) from the larger coordinate (a) and then take the absolute value:
d = | a - b |
Since a > b, the value inside the absolute value function is positive. Therefore, the absolute value of a positive number remains the same. So, we can rewrite the expression as:
d = a - b
In this case, the distance between A and B, as measured on the number line, is a - b. Note that if b was greater than a, we would have gotten -(a - b) instead, which would also simplify to the same value.
So, we can conclude that the distance between two points on a number line, as measured on the x-axis of the coordinate plane, is indeed | a - b |, where a and b are the x-coordinates of the points.