GED Math: The Coordinate Plane

Coordinates let you work with shapes. In a rectangle (or square) with sides parallel to the axes, opposite corners share their x- and y-values, so the missing corner takes the x of one known corner and the y of another. The distance between two points on the same horizontal line is the difference of their x-coordinates; between two points on the same vertical line it is the difference of their y-coordinates (take the positive difference, or count the spaces). The midpoint of a segment is the average of the x-coordinates and the average of the y-coordinates. A triangle whose base lies on a horizontal line has a base equal to that distance, and its height is the vertical distance from the third corner to that line.

1. The Coordinate Plane, Axes, and Ordered Pairs

A coordinate plane (or coordinate grid) is a flat surface used to locate points. It is formed by two number lines that cross at right angles: the x-axis is horizontal and the y-axis is vertical. They meet at the origin, the point (0, 0). Every point is named by an ordered pair (x, y): the first number says how far to move right (positive) or left (negative) along the x-axis, and the second says how far to move up (positive) or down (negative) along the y-axis. Order matters: (2, 3) and (3, 2) are different points. The axes also split the plane into four quadrants, numbered counterclockwise starting at the upper right. Test tip: if a grid is not labeled, the origin is at (0, 0); start there, count left or right first, then up or down.

2. Reading Points from a Grid

To write the coordinates of a point on a grid, start at the origin. First count how many spaces you move horizontally to be directly above or below the point (right is positive, left is negative). Then count how many spaces you move vertically to reach the point (up is positive, down is negative). Write the result as (x, y). Points on the x-axis have y = 0, so they look like (x, 0). Points on the y-axis have x = 0, so they look like (0, y). The origin is the only point that is on both axes.

3. Plotting Points and Using the Quadrants

To plot a point, reverse the reading process: start at the origin, move right or left by the x-coordinate, then up or down by the y-coordinate, and mark the point. The signs of the coordinates tell you the quadrant: Quadrant I (+, +) upper right; Quadrant II (−, +) upper left; Quadrant III (−, −) lower left; Quadrant IV (+, −) lower right. Points on an axis are not in any quadrant. You can also move a point: adding to x moves it right, subtracting moves it left; adding to y moves it up, subtracting moves it down. A reflection across the x-axis changes the sign of y, and a reflection across the y-axis changes the sign of x.

4. Points on a Line

A line on a coordinate grid contains infinitely many points. To decide whether an ordered pair lies on a line, pick two clear points on the line (where it crosses grid intersections), find how the coordinates change as you move from one to the next, and apply the same pattern to test other points. For example, if every step 1 to the right moves the line 2 down, a point 2 to the right of a known point must be 4 lower. The place where the line crosses the y-axis has x = 0 (the y-intercept), and the place where it crosses the x-axis has y = 0 (the x-intercept).

5. Using Coordinates to Find Corners and Distances

Coordinates let you work with shapes. In a rectangle (or square) with sides parallel to the axes, opposite corners share their x- and y-values, so the missing corner takes the x of one known corner and the y of another. The distance between two points on the same horizontal line is the difference of their x-coordinates; between two points on the same vertical line it is the difference of their y-coordinates (take the positive difference, or count the spaces). The midpoint of a segment is the average of the x-coordinates and the average of the y-coordinates. A triangle whose base lies on a horizontal line has a base equal to that distance, and its height is the vertical distance from the third corner to that line.

Common Mistakes to Avoid

• Writing the coordinates in the wrong order. The x-coordinate (horizontal) always comes first.
• Counting up or down first. Always move horizontally first, then vertically.
• Ignoring signs: left of the origin and below the origin are negative.
• Placing a point on an axis into a quadrant. Points with 0 as a coordinate are on an axis.
• Mixing up the quadrant numbers. Quadrant I is upper right, and the numbers increase counterclockwise.
• Miscounting spaces by starting at 1 on the origin. Start counting from the origin as 0.
• Forgetting to use the pattern of a line (the same change each step) when checking a point.
• Mixing up the intercepts: the y-intercept has x = 0, and the x-intercept has y = 0.

Frequently Asked Questions