Distance on the Coordinate Plane

When two points on the coordinate plane sit on the same horizontal or vertical line, you do not need any fancy formula to measure the gap between them. You just count the steps along the grid, or — even faster — subtract the coordinates that change and take the absolute value. The other coordinate stays the same, so it does no work.

Two Points That Share an Axis

• Same x-coordinate (a vertical segment): the points sit one above the other. Distance = |y2 - y1|.
• Same y-coordinate (a horizontal segment): the points sit side by side. Distance = |x2 - x1|.
• Distance is always positive, so we wrap the subtraction in absolute-value bars.
• You can also count the unit squares between the points along the grid line — the answer matches.

Worked Examples

1. A(2, 3) and B(2, 7) share x = 2. Distance = |7 - 3| = 4 units.
2. C(-3, 1) and D(5, 1) share y = 1. Distance = |5 - (-3)| = |5 + 3| = 8 units.
3. E(-4, -2) and F(-4, 6) share x = -4. Distance = |6 - (-2)| = |6 + 2| = 8 units.
4. G(7, -1) and H(-2, -1) share y = -1. Distance = |-2 - 7| = |-9| = 9 units.

Why Absolute Value?

Absolute value is just the distance from zero on a number line, so it is never negative. When the two coordinates straddle zero — like x = -3 and x = 5 — subtraction can come out negative or positive depending on order. Absolute-value bars guarantee the answer is the positive distance, no matter which point you call "first".

When the Points Do NOT Share an Axis

• If neither coordinate matches, the segment is diagonal — counting steps along grid lines no longer works.
• For diagonal segments you need the Pythagorean theorem (you will meet it next year): distance = sqrt((x2-x1)^2 + (y2-y1)^2).
• In Grade 6 you only need to handle horizontal and vertical distances, so always check first whether the x or y values match.