A short history of the coordinate plane
For a long time, algebra (working with numbers and letters) and geometry (working with shapes) were two separate subjects. In the 1600s, the French thinker René Descartes joined them. His idea was simple: give every point two numbers. A popular story says he got the idea while watching a fly on the ceiling and wondering how to describe exactly where it was.
Around the same time, Pierre de Fermat had a similar idea. Because of Descartes (Latin name Cartesius), we call it the Cartesian plane. Much earlier, Indian and Greek mathematicians had already used grids and measured positions on the ground, for example when building altars and temples.
Today the same idea is used in maps, computer screens (every pixel has an (x, y) address), video games and GPS.
2-D coordinates and floor plans
Take two number lines. Lay one flat going left–right: this is the x-axis. Stand the other one up going down–up: this is the y-axis. They cross at right angles at the origin O(0, 0).
- The first number of a point is the x-coordinate (also called the abscissa). It says how far right (+) or left (−) to go.
- The second number is the y-coordinate (the ordinate). It says how far up (+) or down (−) to go.
- (x, y) is an ordered pair: order matters. (3, 2) and (2, 3) are different points.
The four quadrants
The axes cut the plane into four parts called quadrants, numbered anticlockwise from the top right.
| Quadrant | Signs (x, y) | Example |
|---|---|---|
| I | (+, +) | (3, 2) |
| II | (−, +) | (−4, 1) |
| III | (−, −) | (−2, −3) |
| IV | (+, −) | (5, −1) |
A point on an axis is in no quadrant. Points on the x-axis look like (a, 0) and points on the y-axis look like (0, b).
Floor plans
A floor plan is a Cartesian plane drawn to scale. If one square on paper stands for 1 metre, a room that is 5 squares by 4 squares is 5 m by 4 m. Put the origin at one corner of the room and you can give the address of every door, window and cupboard.
Distance between two points
To go from A(x₁, y₁) to B(x₂, y₂), first walk across (the x-gap = x₂ − x₁), then walk up (the y-gap = y₂ − y₁). These two walks meet at a right angle, so AB is the longest side of a right triangle. By Pythagoras:
AB = √[(x₂ − x₁)² + (y₂ − y₁)²]
Special cases: two points on the same horizontal line are |x₂ − x₁| apart; a point (x, y) is √(x² + y²) away from the origin. Since we square the gaps, a negative gap does no harm.
Midpoint of a segment
The midpoint M of AB is the point exactly halfway. Halfway across is the average of the x values, and halfway up is the average of the y values:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Check: AM and MB must be equal. If you know one end A and the midpoint M, the other end is B = (2 × Mx − x₁, 2 × My − y₁).
Collinearity and right-angle checks
Find the three distances AB, BC and AC. Call the largest one the long side.
- Collinear (on one straight line): the two short distances add up exactly to the long one. For example 2√5 + √5 = 3√5.
- Right angle: the squares of the two short sides add up to the square of the long side (converse of Pythagoras). The right angle is at the point opposite the long side.
- Otherwise the points make an ordinary triangle.
Tip: compare squares (AB², BC², AC²) first; they are whole numbers and avoid messy roots.
Try it yourself (practical)
At home: Pick a tiled floor. Put a slipper at one tile corner: that is the origin. Stand at tile (3, 4). Stretch a string straight back to the slipper and count it in tile widths. Predict first: √(9 + 16) = 5 tiles. Then check.
Floor plan: On squared paper, draw your room with 1 square = 50 cm. Write the coordinates of the door and the bed.
In the 3D: go to the last step. Keep A at (−3, −2). Predict where C must go so that A, B(0, 0), C are collinear (hint: try (6, 4)). Then move the sliders to check.
Key formulas and definitions
- Point P(x, y): x = abscissa (across), y = ordinate (up)
- Distance AB = √[(x₂ − x₁)² + (y₂ − y₁)²]
- Distance from origin = √(x² + y²)
- Midpoint M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
- Collinear: AB + BC = AC (long side)
- Right angle at B: AB² + BC² = AC²
Worked examples
1. In which quadrant or on which axis do these lie: (−3, 5), (4, −2), (0, −6)?
(−3, 5): x negative, y positive → quadrant II. (4, −2): x positive, y negative → quadrant IV. (0, −6): x is 0 → on the y-axis (in no quadrant).
2. Find the distance between A(2, 3) and B(8, 11).
x-gap = 8 − 2 = 6. y-gap = 11 − 3 = 8. AB = √(36 + 64) = √100 = 10 units.
3. Find the distance of P(−5, 12) from the origin.
OP = √((−5)² + 12²) = √(25 + 144) = √169 = 13 units.
4. Find the midpoint of A(−6, 4) and B(2, −8).
x = (−6 + 2)/2 = −2. y = (4 + (−8))/2 = −2. M = (−2, −2).
5. M(3, 1) is the midpoint of AB and A = (1, −2). Find B.
(1 + x)/2 = 3 → x = 5. (−2 + y)/2 = 1 → y = 4. B = (5, 4).
6. A room plan uses 1 square = 1 m. The door is at (0, 2) and the window is at (6, 10). How far apart are they?
Gaps: 6 and 8. Distance = √(36 + 64) = 10 squares = 10 m.
7. Are A(1, 2), B(3, 6), C(4, 8) collinear?
AB = √(4 + 16) = √20 = 2√5. BC = √(1 + 4) = √5. AC = √(9 + 36) = √45 = 3√5. AB + BC = 3√5 = AC, so yes, they are collinear.
8. Show that A(1, 1), B(4, 1), C(4, 5) form a right triangle.
AB² = 9, BC² = 16, AC² = 9 + 16 = 25. AB² + BC² = 25 = AC², so the angle at B is 90°. (Also: AB is flat and BC is straight up.)
Common mistakes
- Swapping the order: (2, 5) is 2 across and 5 up, not 5 across and 2 up.
- Forgetting the square root in the distance formula: √25 = 5, not 25.
- Subtracting instead of adding for the midpoint: it is (x₁ + x₂)/2, an average.
- Calling a point like (0, 4) a quadrant I point: points on an axis are in no quadrant.