Geometry on the Grid
Coordinate geometry connects algebra to pictures. Most questions reduce to slope, intercepts, distance, and midpoint.
Slope
Slope measures steepness: m = (y2 - y1) / (x2 - x1). For points (1, 2) and (4, 8), m = (8 - 2)/(4 - 1) = 6/3 = 2.
- •Parallel lines share the same slope.
- •Perpendicular lines have slopes that are negative reciprocals: if one slope is 2, the perpendicular slope is -1/2.
Slope-Intercept Form
Write lines as y = mx + b, where m is slope and b is the y-intercept. The line y = 3x - 4 crosses the y-axis at -4 and rises 3 for every 1 it moves right. To find the x-intercept, set y = 0: 0 = 3x - 4, so x = 4/3.
Distance Formula
Distance between two points is sqrt((x2 - x1)^2 + (y2 - y1)^2). Between (0, 0) and (3, 4): sqrt(9 + 16) = sqrt(25) = 5.
Midpoint Formula
The midpoint is the average of coordinates: ((x1 + x2)/2, (y1 + y2)/2). The midpoint of (2, 4) and (6, 10) is (4, 7).
Writing a Line's Equation
Given slope 2 through (1, 3): start with y = 2x + b, plug in to get 3 = 2(1) + b, so b = 1 and the line is y = 2x + 1.
Key Takeaways
- •Slope = rise over run; parallel equal, perpendicular negative-reciprocal.
- •y = mx + b reveals slope and y-intercept instantly.
- •Distance and midpoint come straight from the coordinate formulas.
- •Find b by plugging a known point into y = mx + b.
