Question

1. use structure suppose a, b, and c are all negative numbers. how do you find the distance between points (a, b) and (a, c)?

Explanation:

Step1: Analyze the coordinates

The two points are \((a,b)\) and \((a,c)\). The \(x -\)coordinates are the same (\(x_1=x_2 = a\)).

Step2: Use the distance formula for vertical distance

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Since \(x_1=x_2=a\), then \(d=\sqrt{(a - a)^2+(c - b)^2}=\vert c - b\vert\) (because \(\sqrt{(c - b)^2}=\vert c - b\vert\)). Also, since \(b\) and \(c\) are negative numbers, we can think of it as subtracting the smaller \(y-\)value from the larger \(y-\)value (in terms of absolute value). For example, if \(b=-m\) and \(c=-n\) (\(m,n>0\)), \(\vert c - b\vert=\vert -n-(-m)\vert=\vert m - n\vert\) which is equivalent to subtracting the smaller of \(b\) and \(c\) (when considering their non - negative counterparts) from the larger one.

Answer:

The distance between the points \((a,b)\) and \((a,c)\) is \(\vert b - c\vert\) (or we can calculate it by subtracting the smaller of the two \(y -\)values (\(b\) and \(c\)) from the larger one. Since \(b\) and \(c\) are negative, we can rewrite it as taking the absolute value of the difference of the two numbers, i.e., if \(b=-x\) and \(c =-y\) (\(x,y>0\)), \(\vert b - c\vert=\vert -x-(-y)\vert=\vert y - x\vert\))