SPI Mathematics · Coordinate Geometry

Coordinates and the Distance Formula

A point is fixed by its x and y coordinates, and the distance between two points is the square root of the sum of the squares of their x and y differences.

All Coordinate Geometry notes

Why this matters

Eleven questions in the bank: eight EASY, two MODERATE and the chapter's only HARD one. Four test the words (origin, abscissa, the equation of an axis); the rest put two points into the distance formula, forwards or backwards. The marks go on a minus sign or a lost second answer.

Concept 1 of 3: Axes, quadrants, abscissa and ordinate

Two number lines cross at right angles at zero. Any point is reached by walking along the x-axis and then up or down: those two moves, written (x, y), are its coordinates.

Definition

The origin O is where the axes meet: (0, 0).
The abscissa is the x-coordinate; the ordinate is the y-coordinate.
Every point on the x-axis has y = 0, so the x-axis is the line y = 0; the y-axis is x = 0.
y = k is a horizontal line, parallel to the x-axis; x = k is a vertical line, parallel to the y-axis.

Term	Meaning	Example
Origin	Where the axes meet	(0, 0)
Abscissa	The x-coordinate	Abscissa of (−3, 5) is −3
Ordinate	The y-coordinate	Ordinate of (−3, 5) is 5
Quadrant I, II, III, IV	Signs (+, +), (−, +), (−, −), (+, −)	(−3, 5) is in quadrant II
Point on the x-axis	y-coordinate is 0	(4, 0)
Point on the y-axis	x-coordinate is 0	(0, −2)
Equation of the x-axis	y = 0	Every point (x, 0)
Equation of the y-axis	x = 0	Every point (0, y)
Line y = k	Parallel to the x-axis, k units from it	y = 3 passes through (1, 3) and (7, 3)
Line x = k	Parallel to the y-axis, k units from it	x = −2 passes through (−2, 0) and (−2, 6)
Distance from the x-axis	The size of y	(6, −4) is 4 units from the x-axis Distance from the x-axis is the size of y, and from the y-axis the size of x: the other coordinate.

Quadrants are counted anticlockwise from the top right.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Coordinate Geometry · Coordinates and the Distance Formula

The graph of y = 5 is a line parallel to the

y = k is parallel to the x-axis, not the y-axis

The line y = 3 keeps y fixed while x runs, so it is horizontal: parallel to the x-axis. It is tempting to pair 'y' with the y-axis, which is the wrong way round.

Drill 3 more on axes, quadrants, abscissa and ordinate

Concept 2 of 3: The distance formula

Join the two points and draw a right triangle under the join: one leg along x, one along y. The join is the hypotenuse, so Pythagoras gives its length.

Definition

Distance between $(x_1, y_1)$ and $(x_2, y_2)$ : $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
Distance of $(x, y)$ from the origin: $\sqrt{x^2 + y^2}$ .
If the two points have the same y, the distance is just the gap between the x values (and the same for the same x).
With sines and cosines, finish with $\sin^2\theta + \cos^2\theta = 1$ .

Distance formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad OP = \sqrt{x^2 + y^2}

Worked example

Find the distance between the points (−2, 3) and (4, −5).

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Coordinate Geometry · Coordinates and the Distance Formula

Distance between the points A

(\cos\theta, \text{Sin}\,\theta)

and B

(\sin\theta, -\cos\theta)

Losing a minus sign

Subtracting a negative coordinate adds:

4 - (-2) = 6

, not 2. A wrong difference still gives a neat-looking square root, so check each difference before squaring.

Stopping before the square root

36 + 64 = 100

is the square of the distance. The distance is

\sqrt{100} = 10

; the 100 is often printed as a wrong option.

Drill 3 more on the distance formula

Concept 3 of 3: Finding an unknown coordinate from a distance

When the distance is given and one coordinate is missing, put the missing one in as a letter, square both sides and solve. A point can sit on either side, so expect two answers.

Definition

Write the distance formula with the unknown, then square both sides to remove the root.
$(y - a)^2 = b^2$ gives $y - a = b$ or $y - a = -b$ : two answers.
A point equidistant from A and B: set its distance to A equal to its distance to B, square both sides, and the squared unknown cancels.
A point on the x-axis is $(x, 0)$ ; on the y-axis, $(0, y)$ .

Equal distances

PA = PB \Rightarrow PA^2 = PB^2

Worked example

The distance between the points (1, 2) and (4, y) is 5 units. Find y.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Coordinate Geometry · Coordinates and the Distance Formula

A line segment is of length 10 units. If the coordinates of its one end are (2, - 3) and the abscissa of the other end is 10, then its ordinate is

Keeping only one root

(y - 2)^2 = 16

gives

y - 2 = \pm 4

, so y is 6 or −2. Options often print both values together; a single value is incomplete.

Leaving the squared unknown in an equidistance question

When you set

PA^2 = PB^2

, the

x^2

(or

y^2

) appears on both sides and cancels, leaving a simple equation. If it does not cancel, a square has been expanded wrongly.

Drill 2 more on finding an unknown coordinate from a distance

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

The distance formula
Distance formula
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad OP = \sqrt{x^2 + y^2}$
Finding an unknown coordinate from a distance
Equal distances
$PA = PB \Rightarrow PA^2 = PB^2$

Reference tables (1)

Axes, quadrants, abscissa and ordinate11 rows

Term	Meaning	Example
Origin	Where the axes meet	(0, 0)
Abscissa	The x-coordinate	Abscissa of (−3, 5) is −3
Ordinate	The y-coordinate	Ordinate of (−3, 5) is 5
Quadrant I, II, III, IV	Signs (+, +), (−, +), (−, −), (+, −)	(−3, 5) is in quadrant II
Point on the x-axis	y-coordinate is 0	(4, 0)
Point on the y-axis	x-coordinate is 0	(0, −2)
Equation of the x-axis	y = 0	Every point (x, 0)
Equation of the y-axis	x = 0	Every point (0, y)
Line y = k	Parallel to the x-axis, k units from it	y = 3 passes through (1, 3) and (7, 3)
Line x = k	Parallel to the y-axis, k units from it	x = −2 passes through (−2, 0) and (−2, 6)
Distance from the x-axis	The size of y	(6, −4) is 4 units from the x-axis Distance from the x-axis is the size of y, and from the y-axis the size of x: the other coordinate.

Quadrants are counted anticlockwise from the top right.

Watch out for (5)

y = k is parallel to the x-axis, not the y-axis
→ Axes, quadrants, abscissa and ordinate
Losing a minus sign
→ The distance formula
Stopping before the square root
→ The distance formula
Keeping only one root
→ Finding an unknown coordinate from a distance
Leaving the squared unknown in an equidistance question
→ Finding an unknown coordinate from a distance

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