Coordinate Geometry
Every point has an (x,y) address. Distance uses Pythagoras. Slope is rise over run.
A. The Cartesian Plane
x-axis and y-axis
The x-axis is the horizontal line. The y-axis is the vertical line. They cross at the origin (0, 0).
ExampleEvery point is written as (x, y). x tells you left/right. y tells you up/down.
The Four Quadrants
The axes divide the plane into four regions. Each quadrant has a different combination of + and − signs.
Signs
Quadrant I: (+, +) → right of y-axis, above x-axis
Quadrant II: (−, +) → left of y-axis, above x-axis
Quadrant III: (−, −) → left of y-axis, below x-axis
Quadrant IV: (+, −) → right of y-axis, below x-axis
Quadrant II: (−, +) → left of y-axis, above x-axis
Quadrant III: (−, −) → left of y-axis, below x-axis
Quadrant IV: (+, −) → right of y-axis, below x-axis
Which quadrant is (−4, 3)?
1x = −4 (negative) and y = 3 (positive)
2Negative x, positive y → Quadrant II
Quadrant II
Memory trick:
Start top-right (I) and go anti-clockwise: I(+,+) → II(−,+) → III(−,−) → IV(+,−).
The Cartesian Plane
B. Distance & Midpoint
Distance Formula
Finds the length between two points. It is just Pythagoras — the x and y differences form the two legs of a right triangle.
Formulad = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint Formula
Finds the exact middle point of a line segment. Average the x-coordinates and average the y-coordinates separately.
FormulaM = ((x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2)
Example 1 — Distance between (1, 2) and (4, 6)
1d = √[(4−1)² + (6−2)²]
2= √[3² + 4²] = √[9 + 16] = √25
Distance = 5 (classic 3-4-5 triangle)
Example 2 — Midpoint of (2, 4) and (8, 10)
1M = ((2+8)÷2 , (4+10)÷2)
2= (10÷2 , 14÷2) = (5, 7)
Midpoint = (5, 7)
Example 3 — Find missing endpoint: M=(3,5), A=(1,2), find B
1(1 + x) ÷ 2 = 3 → 1 + x = 6 → x = 5
2(2 + y) ÷ 2 = 5 → 2 + y = 10 → y = 8
B = (5, 8)
⚠ Common mistakes:
(1) Slope formula — always put y-difference on top: (y₂−y₁)÷(x₂−x₁). Students often flip it.
(2) Midpoint — average x and y separately. Do not add all four numbers together.
(3) Distance — square the differences before adding. √(3+4) ≠ √(3²+4²).
C. Slope (Gradient)
Slope
How steep a line is. Rise ÷ Run. Positive slope goes up left-to-right. Negative slope goes down.
Formulam = (y₂ − y₁) ÷ (x₂ − x₁)
Parallel Lines
Two lines that never meet. They have exactly the same slope.
RuleIf m₁ = m₂ → lines are parallel.
Perpendicular Lines
Two lines that cross at a right angle (90°). Their slopes multiply to −1.
Rulem₁ × m₂ = −1
e.g. slope 2 and slope −½ → 2 × −½ = −1 ✓ perpendicular
e.g. slope 2 and slope −½ → 2 × −½ = −1 ✓ perpendicular
Equation of a Line
y = mx + c where m is slope and c is where the line crosses the y-axis.
Exampley = 3x + 2 → slope = 3, y-intercept = 2
Example 1 — Slope between (2, 3) and (5, 9)
Types of Slope
1m = (y₂−y₁) ÷ (x₂−x₁) = (9−3) ÷ (5−2)
2= 6 ÷ 3 = 2
Slope = 2
Example 2 — Line through (1,3), slope=2. Write equation.
1Use y − y₁ = m(x − x₁): y − 3 = 2(x − 1)
2y − 3 = 2x − 2 → y = 2x + 1
y = 2x + 1
⚡ MCQ Tip: Parallel lines have equal slopes. Perpendicular lines: slopes multiply to −1. Horizontal line → m=0. Vertical line → m=undefined. These four facts appear in almost every coordinate geometry MCQ.
D. Special Lines
Horizontal Line
Runs left-right parallel to the x-axis. Slope = 0.
Exampley = 4 is a horizontal line. Every point on it has y = 4. Slope = 0.
Vertical Line
Runs up-down parallel to the y-axis. Slope is undefined (you divide by zero).
Examplex = 3 is a vertical line. Every point on it has x = 3. Slope = undefined.
Key table: m > 0 → rises left-to-right · m < 0 → falls · m = 0 → horizontal · undefined → vertical · m₁=m₂ → parallel · m₁×m₂=−1 → perpendicular.
Quick MCQ Revision
| Formula | Detail |
|---|---|
Distance | √[(x₂−x₁)²+(y₂−y₁)²] |
Midpoint | ((x₁+x₂)÷2, (y₁+y₂)÷2) |
Slope | (y₂−y₁)÷(x₂−x₁) |
Parallel | m₁ = m₂ |
Perpendicular | m₁ × m₂ = −1 |
| Quadrant signs | I(+,+) II(−,+) III(−,−) IV(+,−) |