Coordinate Geometry — Lines, Slopes, Distance

GRE Quantitative · Axiom Academy

LESSON Coordinate Geometry — Lines, Slopes, Distance Every pair of points hides a slope, a distance, and a line — all three come from the same segment. Take two points, (1, 2) and (3, 8) . Slide from the first to the second: the run is how far you moved right, the rise is how far you moved up, and the slope is just their ratio. Watch the triangle build as the point travels the segment. 2. Distance Is the Same Triangle's Hypotenuse Keep the exact same two points and the exact same rise/run legs from Step 1 — but now look at the straight-line segment connecting them instead of its slope. That segment is the hypotenuse of a right triangle, so the Pythagorean theorem hands you its length directly. Length |x_2 - x_1| — the same run from Step 1. Length |y_2 - y_1| — the same rise from Step 1. The segment itself — its length is the distance between the two points. , so the hypotenuse is a square root. Distance from (1, 2) to (4, 6) : the legs are 4 - 1 = 3 and 6 - 2 = 4 , so . 3. One Point Plus One Slope Builds the Whole Line You don't need two points to pin down a line — one point and the slope are enough, because the slope tells you the direction to extend in both ways from that point. Watch the line grow from a single point at slope 2 , then watch a second line pivot into the one and only slope that meets it at a right angle. Worked example: a line through (1, 3) with slope 2 starts from point-slope y - 3 = 2(x - 1) , which expands to slope-intercept form y = 2x + 1 .

This is the written version of the interactive lesson above. See the full GRE Quantitative course.