Revise: Introduction to Coordinate Geometry
Locate points precisely with coordinates, measure the distance and midpoint between them, and describe straight lines with slope and the equation y = mx + c.
Coordinates are ordered pairs (x, y), measured from the origin (0, 0).
(3, 4) means 3 right, 4 up from the origin.
y = mx + c: m is the slope, c is the y-intercept.
Slope 3 through (2,11) → y = 3x + 5.
The x-axis (horizontal) and y-axis (vertical) cross at the origin, (0, 0).
(3, 4) means 3 right, 4 up.
Quadrant I: (+,+). II: (-,+). III: (-,-). IV: (+,-) — numbered counterclockwise from the top-right.
(-5, 7) is in Quadrant II.
Move horizontally first (x), then vertically (y), starting from the origin.
(-4, 2) is 4 left, 2 up.
d = √[(x₂−x₁)² + (y₂−y₁)²] — the Pythagorean theorem applied to coordinates.
(1,2) to (4,6) → d = 5.
A ratio of m:n splits a segment into m+n equal parts; the midpoint is the 1:1 case.
(0,0) to (8,12) in ratio 1:3 → (2, 3).
m = (y₂−y₁)/(x₂−x₁) stays the same wherever you measure it on a straight line.
(1,2) to (5,14) → m = 3.
y = mx + c describes every point on a non-vertical line.
Slope 3 through (2,11) → y = 3x + 5.
Horizontal: y = constant, slope 0. Vertical: x = constant, slope undefined.
(2,5) and (-6,5) → y = 5.
y-intercept: set x = 0. x-intercept: set y = 0.
y = 2x − 8 → y-intercept (0,−8), x-intercept (4,0).
Perpendicular slopes multiply to −1: m₁ × m₂ = −1.
Slope 2/3 → perpendicular slope −3/2.
GPS, maps, and apps all place things using the same coordinate ideas.
Towers at (0,0) and (6,8) → 10 km apart.