8.2 Coordinate Geometry and Distance
Key Takeaways
- Coordinate geometry on GED Math links measurement to graphs through ordered pairs, slope, distance, and real-world rate questions.
- Plot points as (x, y): move horizontally from the origin first, then vertically, and watch signs for negative coordinates.
- Slope equals rise over run and frequently represents a unit rate such as miles per hour, gallons per minute, or dollars per item.
- The distance formula is the Pythagorean theorem applied on the plane, so it measures straight-line length between two points that do not share a coordinate.
- When two points share an x-value or a y-value, plain subtraction gives the distance and the longer distance formula is unnecessary.
Coordinate Geometry on GED Math
The GED assessment targets include locating points in the coordinate plane, interpreting and computing slope, graphing and writing linear equations, and using slope to solve geometric and rate problems. These show up because a graph can model a map, a wheelchair ramp, a delivery route, a phone-plan cost, or a proportional relationship. Treat the coordinate plane as a measuring system, not a decoration.
An ordered pair is written (x, y). The x-coordinate is horizontal distance from the origin (right is positive, left is negative); the y-coordinate is vertical (up is positive, down is negative). The point (3, -2) is 3 units right and 2 units down, landing in Quadrant IV. The four quadrants run counterclockwise: I (+, +), II (-, +), III (-, -), IV (+, -). GED items may ask which point matches a description, which quadrant holds a point, or what a plotted point means in a real context such as 'the cost of buying zero items.'
Coordinate Tools
| Tool | Use it when... | Formula or move |
|---|---|---|
| Horizontal distance | Points share the same y-value | Subtract the x-values |
| Vertical distance | Points share the same x-value | Subtract the y-values |
| Slope | Steepness or a rate is needed | (change in y) / (change in x) |
| Distance formula | Straight-line length between two non-aligned points | sqrt((x2 - x1)^2 + (y2 - y1)^2) |
| Midpoint | A halfway location is needed | Average the x-values; average the y-values |
The slope and distance formulas both appear on the GED formula sheet, so memorizing them is optional; recognizing which one the prompt needs is the graded skill.
Slope as a Rate
Slope is usually defined as rise over run, but GED problems make it concrete. If a graph plots time on the x-axis and distance on the y-axis, the slope is distance per unit time. If x is the number of tickets and y is total cost, the slope is cost per ticket. A steeper positive line has a larger rate.
Worked example. A water-tank graph passes 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. Slope = 24 / 3 = 8 gallons per minute. Always attach the units, because they explain the answer and rule out trap choices.
Sign and shape carry meaning. A negative slope means y falls as x rises, like a fuel tank emptying as miles increase. A slope of 0 is a horizontal line, so y is constant. An undefined slope is a vertical line, which is not a function of x because one x-value pairs with many y-values.
Slope also connects to the line equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept (the value of y when x = 0). In the ticket example, if each ticket costs 8 dollars and there is a 5-dollar service fee, the cost line is y = 8x + 5: the fee is the starting point at x = 0, and the slope is the per-ticket rate. GED items often ask you to read b as the fixed starting amount and m as the repeating per-unit charge.
Distance and the Pythagorean Theorem
Reach for subtraction first. For A(2, 5) and B(9, 5), the y-values match, so the distance is 9 - 2 = 7 units. For A(-3, 4) and B(-3, -6), the x-values match, so the distance is 4 - (-6) = 10 units. Subtracting a negative is a frequent slip, so write the subtraction out.
When both coordinates change, build a right triangle. For A(1, 2) and B(7, 10), the horizontal leg is 7 - 1 = 6 and the vertical leg is 10 - 2 = 8. The straight-line distance is sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 units. That is the classic 6-8-10 triangle, a scaled 3-4-5. Memorizing the 3-4-5 and 5-12-13 triples lets you confirm distances instantly.
GED Graph Checklist
- Read the axis labels and units before any calculation.
- Decide what the item wants: a point, a rate, a distance, or an interpretation.
- Use subtraction for horizontal or vertical distance whenever a coordinate is shared.
- Use slope for rate of change, never for a total amount.
- Confirm the sign and unit: should the answer be positive, negative, or a unit rate?
This checklist prevents the most common coordinate mistake, which is using every number on the graph just because it is printed. A coordinate item is usually about one relationship: how far apart, how fast, which line is steeper, or what a single point means. Identify that one relationship and ignore the rest.
Comparing rates. When two lines appear together, the steeper line has the larger absolute slope, which on the GED usually means the faster rate, the higher price per unit, or the quicker fill time. If one line is flatter, its rate is smaller even when its starting y-intercept is higher. Reading the intercepts and slopes separately keeps these comparisons straight: the intercept answers 'where does it start?' and the slope answers 'how fast does it change?' A line that starts lower but climbs more steeply will eventually pass a line that starts higher but climbs slowly, and the crossing point is where the two quantities are equal.
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?