Do I have to subtract the first point from the second point?

No. The order doesn't matter when you use absolute value. Whether you calculate |y_2 - y_1| or |y_1 - y_2|, the absolute value makes both equal. For example, |3 - 8| = |−5| = 5 and |8 - 3| = |5| = 5.

What if the two points don't share an x- or y-coordinate?

This lesson covers only points that share one coordinate. If points don't share either coordinate, you would need a different approach (the full distance formula) which you'll learn in later mathematics courses.

Why is distance always positive?

Distance measures how far apart things are. You can't be a negative amount of miles away from something. Absolute value ensures we always get a positive answer, which correctly represents how far apart the points truly are.

How do I know if I should use x-coordinates or y-coordinates?

Check which coordinate is the same for both points. If both points have the same x-coordinate, use the y-coordinates. If both points have the same y-coordinate, use the x-coordinates. If neither is the same, the points don't line up horizontally or vertically.

M6MATH-6.3

Finding Distance Between Points

Learn how to find the distance between two points on a coordinate plane that share the same x- or y-coordinate using absolute value.

What you'll do in this lesson

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What this lesson covers

Have you ever wondered how far apart two locations are on a map? On a coordinate plane, you can use the distance formula for points that are lined up horizontally or vertically. When two points share the same x-coordinate (they're on the same vertical line) or the same y-coordinate (they're on the same horizontal line), finding the distance between them is straightforward using absolute value. In this lesson, you'll learn exactly how to calculate these distances and why absolute value is the key tool you need.

Understanding Vertical and Horizontal Alignment

Two points on a coordinate plane form a distance that's easy to calculate when they line up in a specific way. If two points have the same x-coordinate, they sit on a vertical line — imagine a line going straight up and down. If they have the same y-coordinate, they sit on a horizontal line — imagine a line going straight left and right. For example, the points

(3, 2)

and

(3, 8)

both have x-coordinate 3, so they're on the same vertical line. The points

(−2, 5)

and

(7, 5)

both have y-coordinate 5, so they're on the same horizontal line. When points line up this way, you only need to look at one coordinate to find the distance between them.

Finding Distance Using Absolute Value

Distance is always positive — it measures how far apart things are, never a negative amount. This is why absolute value is perfect for finding distance between points. If two points share the same x-coordinate, subtract their y-coordinates and take the absolute value: distance

= |y_2 - y_1|

. If two points share the same y-coordinate, subtract their x-coordinates and take the absolute value: distance

= |x_2 - x_1|

.

Let's say you have

(2, 3)

and

(2, 9)

. They share x-coordinate 2. The distance is

|9 - 3| = |6| = 6

units. Now try

(−4, 5)

and

(3, 5)

. They share y-coordinate 5. The distance is

|3 - (−4)| = |3 + 4| = |7| = 7

units. Notice that it doesn't matter which point you subtract from which — the absolute value makes the answer positive either way.

Why Absolute Value Matters

Without absolute value, subtracting coordinates could give you a negative number. For instance, if you subtract

3 - 9 = −6

, that looks like the distance is negative 6 units. But distance can't be negative. Taking the absolute value of

−6

gives you

6

units, which makes sense. Absolute value essentially erases the negative sign and gives you the true distance. This works no matter which direction you measure on the coordinate plane. A point 5 units to the right is the same distance as a point 5 units to the left. A point 8 units up is the same distance as a point 8 units down. Absolute value captures this idea perfectly.

Common Mistakes to Avoid

One mistake is forgetting to use absolute value and leaving a negative answer. If you calculate

2 - 7 = −5

and say the distance is −5 units, that's wrong — distance must be positive. Another error is confusing which coordinate to use. Always check: Do the points share an x-coordinate or a y-coordinate? If they share x, look at the y-coordinates. If they share y, look at the x-coordinates. A third mistake is mishandling negative numbers. When subtracting a negative, remember that subtracting a negative is the same as adding:

3 - (−4) = 3 + 4 = 7

. Take the absolute value of your result to stay safe.

Key terms

Distance.: The length of the shortest path between two points, always measured as a positive number or zero.
Absolute Value.: The distance a number is from zero on a number line, written as $|x|$ , always positive or zero.
Coordinate Plane.: A grid formed by a horizontal x-axis and vertical y-axis that intersect at the origin, used to locate points.
x-coordinate.: The first number in an ordered pair $(x, y)$ , telling how far left or right a point is from the origin.
y-coordinate.: The second number in an ordered pair $(x, y)$ , telling how far up or down a point is from the origin.
Vertical Line.: A line that goes straight up and down; all points on a vertical line share the same x-coordinate.
Horizontal Line.: A line that goes straight left and right; all points on a horizontal line share the same y-coordinate.

Worked example

Find the distance between the points

(−3, 2)

and

(−3, 7)

First, check which coordinate is the same. The x-coordinates are both

−3

, so these points lie on a vertical line. When points share an x-coordinate, we find distance using their y-coordinates.

The y-coordinates are 2 and 7. Apply the distance formula for points on a vertical line: distance

= |y_2 - y_1|

.

Substitute the values: distance

= |7 - 2| = |5| = 5

units.

Alternatively, if you subtract the other way: distance

= |2 - 7| = |−5| = 5

units. You get the same answer because absolute value always gives a positive result.

The distance between

(−3, 2)

and

(−3, 7)

is 5 units.

Practice questions

What is the distance between the points

(4, −2)

and

(4, 6)

4 units
6 units
8 units
−2 units

Answer: 8 units

Both points have x-coordinate 4, so they're on a vertical line. Use the y-coordinates: distance

= |6 - (−2)| = |6 + 2| = |8| = 8

units. Remember to add when you subtract a negative number.

Two points share a y-coordinate of 3. One point is at

(−5, 3)

and the other is at

(2, 3)

. Explain which coordinate you use to find the distance between them, and calculate the distance.

Answer: You use the x-coordinates because the points share the same y-coordinate. Distance $= |2 - (−5)| = |2 + 5| = |7| = 7$ units.

When two points share a y-coordinate, they lie on a horizontal line, so you only look at how far apart they are horizontally — that is, their x-coordinates. Subtracting x-coordinates and taking absolute value gives the distance.

Maria plotted two points on a grid. Point A is at

(−8, 4)

and Point B is at

(6, 4)

. Without drawing, find the distance between A and B.

Answer: 14 units

Both points have y-coordinate 4, meaning they're on the same horizontal line. The distance depends only on their x-coordinates: distance

= |6 - (−8)| = |6 + 8| = |14| = 14

units.

FAQ

Do I have to subtract the first point from the second point?: No. The order doesn't matter when you use absolute value. Whether you calculate $|y_2 - y_1|$ or $|y_1 - y_2|$ , the absolute value makes both equal. For example, $|3 - 8| = |−5| = 5$ and $|8 - 3| = |5| = 5$ .
What if the two points don't share an x- or y-coordinate?: This lesson covers only points that share one coordinate. If points don't share either coordinate, you would need a different approach (the full distance formula) which you'll learn in later mathematics courses.
Why is distance always positive?: Distance measures how far apart things are. You can't be a negative amount of miles away from something. Absolute value ensures we always get a positive answer, which correctly represents how far apart the points truly are.
How do I know if I should use x-coordinates or y-coordinates?: Check which coordinate is the same for both points. If both points have the same x-coordinate, use the y-coordinates. If both points have the same y-coordinate, use the x-coordinates. If neither is the same, the points don't line up horizontally or vertically.

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