When two points share a coordinate, the segment between them is horizontal or vertical, and its length comes straight from the other coordinates. You will use absolute value to handle points in different quadrants.
Key idea
If two points have the same y (or the same x), the distance is the gap between the other coordinates. On the same side of 0, subtract their absolute values; on opposite sides, add them.
Why it works
From $x = -3$ to $x = 4$: walk 3 units to reach 0, then 4 more units. That is $3 + 4 = 7$, which is $|-3| + |4|$. From 2 to 6 on the same side, you only walk the extra part: $6 - 2 = 4$. Counting the grid squares gives the same answers.
Worked example: Opposite sides of 0
Find the distance between $(-3, 2)$ and $(4, 2)$.
- Same y, so the segment is horizontal. Use the x-coordinates, $-3$ and $4$.
- They are on opposite sides of 0: $|-3| + |4| = 3 + 4 = 7$.
Answer: 7 units
Worked example: Same side of 0
Find the distance between $(2, -1)$ and $(2, -6)$.
- Same x, so the segment is vertical. Use $-1$ and $-6$.
- Both are below 0: $|-6| - |-1| = 6 - 1 = 5$.
Answer: 5 units
Worked example: On a map
On a town map, the school is at $(-5, 1)$ and the library at $(3, 1)$. Each unit is one block. How far apart are they?
- Same y: use $-5$ and $3$, on opposite sides of 0.
- $5 + 3 = 8$.
Answer: 8 blocks
Common mistake
Subtracting the digits across 0: from $-3$ to 4, "$4 - 3 = 1$". That ignores the 3 units on the negative side. Quick check: count the squares, or ask whether 0 is between the two numbers. If it is, add.