Count the steps

The idea

The distance between (x₁, y) and (x₂, y) is |x₂ – x₁|, and the distance between (x, y₁) and (x, y₂) is |y₂ – y₁|, the absolute value of the difference of the changing coordinate.

Across the y-axis

Two points are on the same horizontal line: A (−3, 2) and B (4, 2). A is left of the y-axis and B is right of it.

How far apart are A and B?

• 1 unit
• 7 units
• 12 units

Slide and count

Slide B along the line through A. The little dots count the steps, and the difference of the changing coordinate is written below.

Subtract, then drop the sign

The distance between (x₁, y) and (x₂, y) is |x₂ − x₁|. The distance between (x, y₁) and (x, y₂) is |y₂ − y₁|. It is the absolute value of the difference of the coordinate that changes.

• Points | Distance
• (−3, 2) and (4, 2): x changes | |4 − (−3)| = |7| = 7
• (−3, 2) and (−7, 2): x changes | |−7 − (−3)| = |−4| = 4
• (−3, 2) and (−3, −1): y changes | |−1 − 2| = |−3| = 3

Notes

Distance along a line parallel to an axis: |x₂ − x₁| (same y) or |y₂ − y₁| (same x).

Check yourself

What is the distance between (2, 5) and (9, 5)?

Answer: 7

Only x changes: |9 − 2| = 7.

What is the distance between (−4, 1) and (−4, −5)?

Answer: 6

Only y changes: |−5 − 1| = |−6| = 6.

Which expression gives the distance between (6, −2) and (−1, −2)?

A (−6, 3) and B (2, 3). How far is A from B?

Answer: 8

|2 − (−6)| = |8| = 8.

• |−1 − 6| — correct. Yes. The y-coordinates are equal, so only x changes: |−1 − 6| = |−7| = 7.
• |−2 − (−2)|. That is 0. The y-coordinates are the same and do not change; the x-coordinates are the ones that differ.
• |6 + (−1)|. That adds the x-coordinates. Subtract them.