Skill: Determine horizontal and vertical distances on a coordinate plane

To find the horizontal distance between two points, compare their $x$-coordinates:

$$\text{horizontal distance}=\left \vert x_2-x_1\right \vert$$

To find the vertical distance, compare their $y$-coordinates:

$$\text{vertical distance}=\left \vert y_2-y_1\right \vert$$

The absolute value symbols $\left \vert \right \vert$ make the distance nonnegative, even if the points are in different quadrants or the coordinates are decreasing.

Example

Find the horizontal and vertical distances between $A(-3,4)$ and $B(5,-2)$.

Step 1: Find the horizontal distance.

Use the $x$-coordinates, $-3$ and $5$:

$$\left \vert 5-(-3)\right \vert =\left \vert 8\right \vert =8$$

The horizontal distance is 8 coordinate units.

Step 2: Find the vertical distance.

Use the $y$-coordinates, $4$ and $-2$:

$$\left \vert -2-4\right \vert =\left \vert -6\right \vert =6$$

The vertical distance is 6 coordinate units.

So, the points are 8 units apart horizontally and 6 units apart vertically. These measurements describe movement left or right and up or down; they do not measure the diagonal distance between the points.