Coordinate geometry combines algebra and geometry using coordinates to describe positions and relationships.
x-axis: Horizontal line
y-axis: Vertical line
Origin: Point (0, 0) where axes meet
Coordinates: (x, y) where x is horizontal, y is vertical
I: (+x, +y) top right
II: (-x, +y) top left
III: (-x, -y) bottom left
IV: (+x, -y) bottom right
Find the distance between any two points using the distance formula.
d = √[(x₂-x₁)² + (y₂-y₁)²]
Distance between points (x₁, y₁) and (x₂, y₂)
Find distance between (1, 2) and (4, 6)
d = √[(4-1)² + (6-2)²]
d = √[3² + 4²] = √[9 + 16] = √25 = 5
Find the point exactly halfway between two given points.
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Midpoint between (x₁, y₁) and (x₂, y₂)
Find midpoint of (2, 4) and (8, 10)
M = ((2+8)/2, (4+10)/2) = (10/2, 14/2) = (5, 7)
Gradient measures how steep a line is - the rate of change of y with respect to x.
m = (y₂-y₁)/(x₂-x₁)
Gradient between points (x₁, y₁) and (x₂, y₂)
Line slopes upward ↗
m > 0
Line slopes downward ↘
m < 0
Horizontal line →
m = 0
There are several ways to write the equation of a straight line.
y = mx + c
m = gradient
c = y-intercept
Most common form
y - y₁ = m(x - x₁)
When you know gradient and one point
Useful for finding equations
1. Find the distance between points (0, 0) and (3, 4)
2. Find the gradient of the line through points (1, 2) and (5, 10)