8.2 Coordinate Geometry and Distance
Key Takeaways
- Coordinate geometry on GED Math connects measurement with graphs, slope, distance, and real-world rate questions.
- Plot points as ordered pairs in the form (x, y), moving horizontally first and vertically second.
- Slope is rise over run and can represent a unit rate, such as miles per hour or dollars per item.
- The distance formula is a coordinate-plane version of the Pythagorean theorem, so it measures straight-line length between two points.
- Horizontal and vertical distances can often be found by subtraction before using a more complicated formula.
Coordinate Geometry on GED Math
The GED assessment targets include locating points in the coordinate plane, interpreting slope, graphing linear equations, and using slope to solve geometric problems. These skills overlap with geometry because a graph can represent a map, a ramp, a delivery route, a cost pattern, or a proportional relationship. Treat the coordinate plane as a measuring system, not just a picture.
An ordered pair is written as (x, y). The x-coordinate tells how far to move left or right from the origin, and the y-coordinate tells how far to move up or down. For example, the point (3, -2) is 3 units right and 2 units down. GED questions may ask which point matches a description, which quadrant contains a point, or what a point means in context.
Coordinate Tools
| Tool | Use it when... | Formula or move |
|---|---|---|
| Horizontal distance | Points have the same y-value | Subtract x-values |
| Vertical distance | Points have the same x-value | Subtract y-values |
| Slope | A line's steepness or rate is needed | (change in y) / (change in x) |
| Distance formula | Straight-line distance between two non-aligned points is needed | sqrt((x2 - x1)^2 + (y2 - y1)^2) |
| Midpoint idea | A halfway location is needed | Average the x-values and average the y-values |
Slope as Rate
Slope is often written as rise over run, but GED problems usually make it practical. If a graph has time on the x-axis and distance on the y-axis, slope is distance per unit of time. If x is number of tickets and y is total cost, slope is cost per ticket. A steeper positive line has a larger rate.
Suppose a water tank graph goes through (2, 18) and (5, 42), where x is minutes and y is gallons. The change in y is 42 - 18 = 24 gallons. The change in x is 5 - 2 = 3 minutes. The slope is 24 / 3 = 8 gallons per minute. The units matter because they explain the answer.
A negative slope means y decreases as x increases. If a car's fuel tank has a line with negative slope, the amount of gas is going down as miles increase. A slope of 0 means the line is horizontal and the y-value is constant. An undefined slope means the line is vertical, which usually does not represent a function of x because one x-value has many y-values.
Distance and the Pythagorean Theorem
Use simple subtraction first. If two points are A(2, 5) and B(9, 5), the y-values match, so the distance is 9 - 2 = 7 units. If the points are A(-3, 4) and B(-3, -6), the x-values match, so the distance is 4 - (-6) = 10 units.
When both coordinates change, make a right triangle. For A(1, 2) and B(7, 10), the horizontal change is 7 - 1 = 6 and the vertical change is 10 - 2 = 8. The straight-line distance is sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 units. This is the same 6-8-10 right triangle that the Pythagorean theorem would produce.
GED Graph Checklist
- Read axis labels and units before doing calculations.
- Identify whether the question asks for a point, a rate, a distance, or an interpretation.
- Use subtraction for horizontal or vertical distance when possible.
- Use slope for rate of change, not for total amount.
- Check whether the answer should be positive, negative, or a unit rate.
This checklist prevents a common GED mistake: using every number in the graph just because it appears. A coordinate problem is usually about one relationship, such as how far apart, how fast, which line is steeper, or what a point means.
A line on a GED Math graph passes through (1, 6) and (5, 18). What is the slope of the line?
What is the distance between points A(2, 3) and B(8, 11) on the coordinate plane?