Distance Between Points That Share a Coordinate

How far apart are two points on the coordinate plane? In this lesson you will find out. Start with what you already know about an ordered pair $(x, y)$ : it is two signed numbers (each can be positive, negative or zero), called coordinates, in a fixed order. The first coordinate, $x$ , is the horizontal position. A positive $x$ puts the point to the right of the vertical axis, and a negative $x$ puts it to the left. The second coordinate, $y$ , is the vertical position. A positive $y$ puts the point above the horizontal axis, and a negative $y$ puts it below.

The distance between two points is the length of the straight path that joins them. You will measure it in units, where $1$ unit is the distance from $0$ to $1$ on either axis.

In this lesson you will only measure between two points that share a coordinate. That means the two points have the same $x$ or the same $y$ . The method you are about to learn depends on that.

Why does a shared coordinate help? Look at the points $(2, 5)$ and $(9, 5)$ . Both have $y = 5$ , so they are at the same height, and the straight path between them runs left to right. Only the horizontal positions, $2$ and $9$ , are different.

So you can set the shared $5$ aside and treat $2$ and $9$ as two positions on one number line. Counting the one-unit steps from $2$ to $9$ gives $7$ , so the two points are $7$ units apart.

The same idea works when the $x$ -coordinates match. Then one point is directly above the other, the path runs straight up and down, and you compare the two $y$ -coordinates on one number line.

Every problem in this lesson comes down to one question: how far apart are two positions on a number line?

To measure on a number line, you use absolute value. The absolute value of a number $a$ , written $\lvert a \rvert$ , is the distance from $a$ to $0$ . A distance is never negative, so $\lvert a \rvert$ is never negative. For example, $\lvert 8 \rvert = 8$ and $\lvert -8 \rvert = 8$ , because $8$ and $-8$ are both $8$ units from $0$ . Also, $\lvert 0 \rvert = 0$ .

Do not confuse absolute value with order. Order follows position on the line: the number farther left on a horizontal line, or lower on a vertical line, is the lesser number. So $-8$ is less than $-3$ . But $-8$ is farther from $0$ than $-3$ is, so $-8$ has the larger absolute value: $\lvert -8 \rvert = 8$ and $\lvert -3 \rvert = 3$ . In this lesson, larger absolute value always means farther from $0$ , not greater in order.

Now picture moving along the number line from one position to the other, and ask: do you pass through $0$ ?

Same side of 0 (both positive or both negative): you do not pass through $0$ . Both positions lie in the same direction from $0$ , so the nearer one sits on the path from $0$ to the farther one. The gap between them is the larger absolute value minus the smaller absolute value. For $2$ and $9$ , that is $9 - 2 = 7$ , the same $7$ units you counted.
Opposite sides of 0 (one negative, one positive): you do pass through $0$ . You cover the first position's distance from $0$ and then the second's, so the gap is the two absolute values added together.

If one position is $0$ itself, the gap is just the absolute value of the other position.

The table tests this reasoning. Four rows use positions that are $3$ units and $8$ units from $0$ , and the last row includes $0$ itself.

Where two positions sit decides whether you subtract or add absolute values

Positions	Where they sit	Reasoning	Distance apart
$3$ and $8$	Same side of $0$ (both positive)	$\lvert 8 \rvert - \lvert 3 \rvert = 8 - 3 = 5$	$5$ units
$-3$ and $-8$	Same side of $0$ (both negative)	$\lvert -8 \rvert - \lvert -3 \rvert = 8 - 3 = 5$	$5$ units
$-3$ and $8$	Opposite sides of $0$	$\lvert -3 \rvert + \lvert 8 \rvert = 3 + 8 = 11$	$11$ units
$3$ and $-8$	Opposite sides of $0$	$\lvert 3 \rvert + \lvert -8 \rvert = 3 + 8 = 11$	$11$ units
$0$ and $-8$	One position is $0$	$\lvert -8 \rvert = 8$	$8$ units

Let $p$ and $q$ be two positions on a number line. There is a short way to write the distance between them: $\lvert p - q \rvert$ . This is called the absolute difference of $p$ and $q$ . Like every absolute value, it is never negative. It is also the same whichever position you call $p$ , because the gap is the same in both directions.

When both positions are positive, you can check it by subtracting the smaller from the larger: $\lvert 8 - 3 \rvert = \lvert 5 \rvert = 5$ , which matches the first row of the table. When a position is negative, you do not need a rule for subtracting negative numbers. The same-side and opposite-side reasoning is how you find the value. For example, $\lvert 8 - (-3) \rvert$ stands for the distance between $8$ and $-3$ , and the third row shows that this distance is $3 + 8 = 11$ units.

Key concept

Use this when two points share a coordinate (same $x$ or same $y$ ). Let $p$ and $q$ be the two coordinates that differ, and treat them as positions on one number line. The distance between the points is $\lvert p - q \rvert$ . Find it from $\lvert p \rvert$ and $\lvert q \rvert$ , the distances of $p$ and $q$ from $0$ , which are never negative.

Same side of 0: larger absolute value minus smaller absolute value.
Opposite sides of 0: add the two absolute values.
One position is 0: the absolute value of the other position.

Choose larger by distance from $0$ , not by order. Order follows position on the line, so $-8$ is less than $-3$ , yet $\lvert -8 \rvert$ is the larger absolute value.

\text{distance between } p \text{ and } q = \lvert p - q \rvert

Worked example

Both positions on the same side of 0

Find the distance between the points $(-7, 3)$ and $(-2, 3)$ .

Look for the shared coordinate. Both points have y-coordinate 3, so they are at the same height and the path between them is horizontal. Compare the x-coordinates as positions on one number line, and call them p and q.
$p = -7, \quad q = -2$
Decide where the positions sit. Both are negative, so they are on the same side of 0, and the path between them does not pass through 0.
Find each position's distance from 0. This is its absolute value, which is never negative.
$\lvert -7 \rvert = 7, \quad \lvert -2 \rvert = 2$
Because the positions are on the same side of 0, the nearer one sits on the path from 0 to the farther one, so subtract the smaller absolute value from the larger. Negative 7 is the lesser number because it is farther left, yet it has the larger absolute value. The result is the absolute difference of p and q.
$\lvert p - q \rvert = \lvert -7 \rvert - \lvert -2 \rvert = 7 - 2 = 5$

Answer

The points $(-7, 3)$ and $(-2, 3)$ are $5$ units apart.

Worked example

Positions on opposite sides of 0

Find the distance between the points $(4, -6)$ and $(4, 5)$ .

Both points have x-coordinate 4, so one is directly above the other and the path between them is vertical. Compare the y-coordinates as positions on one number line, and call them p and q.
$p = -6, \quad q = 5$
One position is negative and the other is positive, so they are on opposite sides of 0. Find each position's distance from 0, its absolute value.
$\lvert -6 \rvert = 6, \quad \lvert 5 \rvert = 5$
Because the positions are on opposite sides of 0, the path between them passes through 0: 6 units to reach 0, then 5 more units. Add the two absolute values to get the absolute difference of p and q.
$\lvert p - q \rvert = \lvert -6 \rvert + \lvert 5 \rvert = 6 + 5 = 11$

Answer

The points $(4, -6)$ and $(4, 5)$ are $11$ units apart.

Worked example

Two distances, two different cases

Point $A$ is at $(-6, -2)$ , point $B$ is at $(-6, -9)$ and point $C$ is at $(3, -9)$ . Find the distance from $A$ to $B$ and the distance from $B$ to $C$ .

Points A and B have the same x-coordinate, negative 6, so compare their y-coordinates. The shared coordinate only shows that the path is vertical; it is not part of the arithmetic. Both y-coordinates are negative, so the positions are on the same side of 0.
$-2 \text{ and } {-9}$
On the same side of 0, the path between the positions does not pass through 0, so subtract the smaller absolute value from the larger.
$\lvert -9 \rvert - \lvert -2 \rvert = 9 - 2 = 7$
Points B and C have the same y-coordinate, negative 9, so compare their x-coordinates. One is negative and the other is positive, so the positions are on opposite sides of 0.
$-6 \text{ and } 3$
On opposite sides of 0, the path between the positions passes through 0, so add the two absolute values.
$\lvert -6 \rvert + \lvert 3 \rvert = 6 + 3 = 9$

Answer

The distance from $A$ to $B$ is $7$ units, and the distance from $B$ to $C$ is $9$ units.

The takeaway: When two points share a coordinate, the distance between them is the gap between the two coordinates that differ, read as positions $p$ and $q$ on one number line. That gap is $\lvert p - q \rvert$ , and it is never negative. If the positions are on the same side of $0$ , subtract the smaller absolute value from the larger one, where larger means farther from $0$ , not greater in order. If they are on opposite sides of $0$ , add the two absolute values. Next, you will graph ordered pairs from a table of values and interpret points on a coordinate plane in real-world situations.