Coordinate Geometry Basics: Horizontal and Vertical Movement on the Plane

1) Coordinate Changes as Movement

When you move from one point to another on the coordinate plane, you can describe the movement by how x and y change. A key idea is that holding one coordinate constant forces the movement to go in a single direction.

Horizontal movement: keep y constant

If two points have the same y-value, the segment connecting them is horizontal. That means you move left or right only.

• Same y (for example, y = 3 for both points)  horizontal segment
• x changes  movement is left/right

Example: From A(-2, 3) to B(5, 3), the y-value stays 3, so the segment is horizontal.

Vertical movement: keep x constant

If two points have the same x-value, the segment connecting them is vertical. That means you move up or down only.

• Same x (for example, x = -4 for both points)  vertical segment
• y changes  movement is up/down

Example: From C(1, -5) to D(1, 2), the x-value stays 1, so the segment is vertical.

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Neither horizontal nor vertical

If both x and y change between the endpoints, the segment is neither horizontal nor vertical (it is slanted/diagonal).

Example: From E(-1, 0) to F(3, 4), x changes and y changes, so it is neither.

2) Computing Horizontal and Vertical Distance (Absolute Difference)

For horizontal and vertical segments, the length is found by subtracting the coordinates that change and taking the absolute value (so the distance is never negative).

Horizontal distance formula

If A(x1, y) and B(x2, y) share the same y, then the horizontal distance is:

distance = |x2 - x1|

Step-by-step example: Find the length of segment AB where A(-2, 3) and B(5, 3).

• Check: y-values match (3 and 3)  horizontal
• Subtract x-coordinates: 5 - (-2) = 7
• Absolute value: |7| = 7
• So, AB = 7 units

Vertical distance formula

If C(x, y1) and D(x, y2) share the same x, then the vertical distance is:

distance = |y2 - y1|

Step-by-step example: Find the length of segment CD where C(1, -5) and D(1, 2).

• Check: x-values match (1 and 1)  vertical
• Subtract y-coordinates: 2 - (-5) = 7
• Absolute value: |7| = 7
• So, CD = 7 units

Why absolute value?

Distance measures how far apart points are, not direction. For instance, -5 - 2 = -7, but the distance is still 7 units, so we use |-7| = 7.

Quick reference table

3) Practice Set: Classify Segments from Endpoint Coordinates

For each pair of endpoints, decide whether the segment is horizontal, vertical, or neither. If it is horizontal or vertical, compute its length using absolute difference.

Problems

• 1. A(4, -1) to B(-3, -1)
• 2. C(2, 6) to D(2, -4)
• 3. E(-5, 0) to F(1, 3)
• 4. G(0, 7) to H(9, 7)
• 5. I(-2, -2) to J(-2, 5)
• 6. K(3, 1) to L(-1, 1)
• 7. M(-4, 2) to N(-4, 2)
• 8. P(6, -3) to Q(1, -8)

Answer check (classification + length when applicable)

• 1. Horizontal (same y=-1), length |-3 - 4| = 7
• 2. Vertical (same x=2), length |-4 - 6| = 10
• 3. Neither (both coordinates change)
• 4. Horizontal (same y=7), length |9 - 0| = 9
• 5. Vertical (same x=-2), length |5 - (-2)| = 7
• 6. Horizontal (same y=1), length |-1 - 3| = 4
• 7. Horizontal and vertical at the same time (same point), length 0
• 8. Neither (both coordinates change)

4) Visual Check: Sketch and Confirm Lengths on the Grid

Now verify your computations by drawing. This helps you connect the subtraction method to actual grid spacing.

Sketching steps (use for any problem)

• Step 1: Plot the two endpoints on a coordinate grid.
• Step 2: Draw the segment connecting the points.
• Step 3: Decide if the segment is horizontal or vertical by checking whether it lies along a grid line (perfectly left-right or up-down).
• Step 4: Count grid units between endpoints in the direction of movement (left-right for horizontal, up-down for vertical).
• Step 5: Compare the counted units to your computed absolute difference.

Guided visual checks

Check A: Use problem 1: (4, -1) to (-3, -1).

• The segment should lie on the horizontal line y=-1.
• Counting from x=-3 to x=4 gives 7 unit steps, matching |-3-4|=7.

Check B: Use problem 2: (2, 6) to (2, -4).

• The segment should lie on the vertical line x=2.
• Counting from y=-4 to y=6 gives 10 unit steps, matching |-4-6|=10.

Check C (spot the difference): Use problem 3: (-5, 0) to (1, 3).

• The segment will be slanted, not along a single grid line.
• Notice that counting only horizontal change (|1-(-5)|=6) or only vertical change (|3-0|=3) does not give the segment length, because the movement is in two directions at once.