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# Coordinate Geometry Study Guide: GED Math

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Updated on Mar 23, 2011

Practice problems for these concepts can be found at:

Geometry Practice Problems: GED Math

### Coordinate Geometry

The coordinate plane is the grid of boxes on which the x-axis and y-axis are placed and coordinate points called ordered pairs can be plotted. The points and figures that can be plotted on the plane and the operations that can be performed on them fall under the heading coordinate geometry.

The graph is an example of the coordinate plane. The x-axis and the y-axis meet at the origin, a point with the coordinates (0,0). The origin is zero units from the x-axis and zero units from the y-axis.

Look at the point labeled A, with the coordinates (2,3). The x-coordinate is listed first in the coordinate pair. The x value of a point is the distance from the y-axis to that point. Point A is two units from the y-axis, so its x value is 2. The y-coordinate is listed second in the coordinate pair. The y value of a point is the distance from the x-axis to that point. Point A is three units from the x-axis, so its y value is 3.

What are the coordinates of point B? Point B is minus four units from the y-axis and four units from the x-axis. The coordinates of point B are (–4,4).

The coordinate plane is divided into four sections, or quadrants. The points in quadrant I, the top right corner of the plane, have positive values for both x and y. Point A is in quadrant I and its x and y values are both positive. The points in quadrant II, the top left corner of the plane, have negative values for x and positive values for y. Point B is in quadrant II and its x value is negative, while its y value is positive. The points in quadrant III, the bottom left corner of the plane, have negative values for both x and y, and the points in quadrant IV, the bottom right corner of the plane, have positive values for x and negative values for y.

### Slope

When two points on the coordinate plane are connected, a line is formed. The slope of a line is the difference between the y values of two points divided by the difference between the x values of those two points. When the equation of a line is written in the form y = mx + b, the value of m is the slope of the line.

If both the y value and the x value increase from one point to another, or if both the y value and the x value decrease from one point to another, the slope of the line is positive. If the y value increases and the x value decreases from one point to another, or if the y value decreases and the x value increases from one point to another, the slope of the line is negative.

A horizontal line has a slope of 0. Lines such as y = 3, y = –2, or y = c, where c is any constant, are lines with slopes of 0.

A vertical line has no slope. Lines such as x = 3, x = –2, or x = c are lines with no slopes.

The slope of the line AB at left is equal to [5 – (–3)] ÷ [2 – (–2)] = = 2. The slope of line AB is 2. The slope of line EF is equal to [(–5) – 4] ÷ (9 – 6) = = –3. Line GH is a horizontal line; there is no change in the y values from point G to point H. This line has a slope of 0. Line CD is a vertical line; there is no change in the x values from point C to point D. This line has no slope.

Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other. Lines given by the equations y = 3x + 5 and y = 3x – 2 are parallel, while the line given by the equation y = –x + 1 is perpendicular to those lines.

### Midpoint

The midpoint of a line segment is the coordinates of the point that falls exactly in the middle of the line segment. If (a,c) is one endpoint of a line segment and (b,d) is the other endpoint, the midpoint of the line segment is equal to . In other words, the midpoint of a line segment is equal to the average of the x values of the endpoints and the average of the y values of the endpoints.

Example

What is the midpoint of a line segment with endpoints at (1,5) and (–3,3)?

Using the midpoint formula, the midpoint of this line is equal to .

### Distance

To find the distance between two points, use the following formula. The variable x1 represents the x-coordinate of the first point, x2 represents the x coordinate of the second point, y1 represents the y-coordinate of the first point, and y2 represents the y-coordinate of the second point:

Example

What is the distance between the points (–2,8) and (4,–2)?

Substitute these values into the formula:

Practice problems for these concepts can be found at:

Geometry Practice Problems: GED Math