Points on the Plane: Coordinate Geometry
Coordinate geometry (also called analytic geometry) uses the x-y coordinate plane to describe the position, distance, and relationships of points, lines, and shapes using algebraic formulas. By assigning coordinates to geometric objects, students can calculate distances, find midpoints, determine slopes, and prove geometric properties through computation.
Components of Coordinate Geometry
This section covers the essential formulas and concepts:
- Plotting Points & Quadrants: Every point is identified by an ordered pair (x, y), placed in one of four quadrants or on an axis.
- Distance Formula: The distance between two points (x₁, y₁) and (x₂, y₂) is d = √((x₂ - x₁)² + (y₂ - y₁)²).
- Midpoint Formula: The midpoint of a segment is ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Slope: The steepness of a line between two points is m = (y₂ - y₁) / (x₂ - x₁), describing rise over run.
Examples of Coordinate Geometry
Plotting Points Examples
- The point (3, -2) is in Quadrant IV: 3 units right and 2 units down from the origin.
- The point (-4, 5) is in Quadrant II: 4 units left and 5 units up from the origin.
- The point (0, 6) lies on the y-axis, not in any quadrant.
Distance Formula Examples
- Find the distance between (1, 2) and (4, 6): d = √((4-1)² + (6-2)²) = √(9 + 16) = √25 = 5.
- Find the distance between (-3, 1) and (5, 1): d = √((5-(-3))² + (1-1)²) = √(64 + 0) = 8 (a horizontal segment).
- Two towns are at coordinates (2, 3) and (10, 9) on a map where each unit is 1 mile. Distance = √(64 + 36) = √100 = 10 miles.
Midpoint Formula Examples
- Find the midpoint of (2, 8) and (6, 4): M = ((2+6)/2, (8+4)/2) = (4, 6).
- Find the midpoint of (-3, 5) and (7, -1): M = ((-3+7)/2, (5+(-1))/2) = (2, 2).
- A fence runs from (0, 0) to (12, 8). The center post goes at the midpoint: (6, 4).
Slope Examples
- Find the slope between (1, 3) and (4, 9): m = (9-3)/(4-1) = 6/3 = 2.
- Find the slope between (2, 7) and (5, 7): m = (7-7)/(5-2) = 0/3 = 0 (a horizontal line).
- Find the slope between (-1, 4) and (3, -2): m = (-2-4)/(3-(-1)) = -6/4 = -3/2 (a downward line from left to right).