y = mx + b
Fast when slope and y-intercept are known.
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GMAT Quant
Coordinate Geometry
Use slope, line equations, midpoint, distance, and parallel or perpendicular relationships efficiently.
Slope
Slope measures vertical change per horizontal change.
For two points, m = (y2 - y1)/(x2 - x1).
Recognition
Recognize slope from the line.
- 01Positive slope rises left to right.
- 02Negative slope falls left to right.
- 03Zero slope is horizontal.
- 04Undefined slope is vertical.
Example: points (1, 2) and (5, 10)
- Rise
- 10 - 2 = 8
- Run
- 5 - 1 = 4
- Slope
- 8/4 = 2
Use the line form that matches the information.
y - y1 = m(x - x1)
Fast when slope and one point are known.
Midpoint averages coordinates.
- Points
- (2, 5) and (8, 1)
- x midpoint
- (2 + 8)/2 = 5
- y midpoint
- (5 + 1)/2 = 3
- Midpoint
- (5, 3)
Distance
Distance is the Pythagorean theorem in coordinates.
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2). For purely horizontal or vertical distances, just subtract coordinates.
Parallel and perpendicular lines use slope differently.
Parallel
Same slope.
Perpendicular
For finite nonzero slopes, slopes are negative reciprocals, so m1*m2 = -1.
Mini application
Try this: a line has slope 4 and passes through (3, 1).
Write a point-slope equation before scrolling.
Solution
One correct form is y - 1 = 4(x - 3).
You can expand it to y = 4x - 11 only if the question needs slope-intercept form.
Remember this
Match the shortcut to the geometry.
Slope for direction, point-slope for a known point, midpoint for centers, and simple subtraction for horizontal or vertical distance.