Learn: Coordinate Geometry

Review the key ideas, then practise whenever you’re ready.

The idea

Coordinate geometry places points on an $(x, y)$ grid so we can measure shapes with simple arithmetic. Drawing the horizontal run and vertical rise between two points $(x_1, y_1)$ and $(x_2, y_2)$ makes a right triangle - and that triangle gives you gradient, midpoint, and distance.

The four coordinate tools:

1
Gradient / Slope ( $m$ ) - $m = \frac{y_2 - y_1}{x_2 - x_1}$ - vertical rise divided by horizontal run.
2
Midpoint ( $M$ ) - $M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)$ - average the $x$ ’s and average the $y$ ’s.
3
Distance ( $d$ ) - $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ - Pythagoras on the horizontal run and vertical rise.
4
Line Equation ( $y = mx + c$ ) - $m$ is gradient, $c$ is $y$ -intercept (at $x = 0$ ). Parallel lines share $m$ ; perpendicular slopes multiply to $-1$ .

Quick memory tips

You Rise out of bed before you Run outside - vertical change $y$ always goes on top of horizontal change $x$
Slope and Distance subtract coordinates, but Midpoint is an average so it ADDS them
Perpendicular ( $90^\circ$ ) slope = flip the fraction upside down AND switch its sign

Interactive Coordinate Triangle Explorer

Every two points A(1, 1) and B(5, 4) form a right-angled triangle on the grid.

Gradient m = Rise ÷ Run

From A(1, 1) to B(5, 4), the line steps 4 units right (run) and climbs 3 units up (rise). Always put vertical $y$ on top!

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{5 - 1} = \frac{3}{4}

Worked example

Every step explained - why we do it, and what happens.

1
$A(1, 2) \quad \text{and} \quad B(4, 11)$
Why: We want the straight-line equation y = mx + c through A(1, 2) and B(4, 11). First find the gradient m, then find the y-intercept c.
What happens: Two points give us the rise and the run.
2
$\text{Rise} = 11 - 2 = 9, \qquad \text{Run} = 4 - 1 = 3$
Why: From A to B, y climbs from 2 to 11 (rise = 9) while x steps from 1 to 4 (run = 3).
What happens: Both legs of the coordinate triangle are ready.
3
$m = \frac{9}{3} = 3$
Why: Divide vertical rise (9) by horizontal run (3).
What happens: The gradient is m = 3, so the line looks like y = 3x + c.
4
$2 = 3(1) + c \;\Rightarrow\; c = -1$
Why: Substitute A(1, 2) into y = 3x + c and subtract 3 from both sides.
What happens: The y-intercept is c = −1, giving y = 3x − 1.
5
$3(4) - 1 = 12 - 1 = 11$
Why: The check: test the second point B(4, 11) in y = 3x − 1.
What happens: At x = 4, y = 11 ✓ - the line y = 3x − 1 passes through both points.

Intercepts, Parallel & Perpendicular Lines (Optional Toolbox)

How to read $y = mx + c$ , find axis crossings, and spot parallel or $90^\circ$ perpendicular slopes.

Levels 3–6

1. Axis Intercepts

y-intercept: set $x = 0$ (gives $c$ in $y = mx + c$ ).
x-intercept: set $y = 0$ and solve for $x$ .

2. Parallel Lines (∥)

Parallel lines never cross because they climb at the exact same steepness: $m_1 = m_2$ .

3. Perpendicular Lines (⊥ at 90°)

Lines meeting at $90^\circ$ have negative reciprocal slopes ( $m_1 \cdot m_2 = -1$ ): flip the fraction upside down and switch its sign (for example, $\frac{2}{3}$ becomes $-\frac{3}{2}$ ).

Real-world example

On a rescue grid (in km), a helicopter base is at A(−2, 1) and a hiker is at B(4, 9). How far is the straight-line flight, and where is the halfway radio relay M?

1
$\Delta x = 4 - (-2) = 6 \text{ km}, \qquad \Delta y = 9 - 1 = 8 \text{ km}$
Why: Find the horizontal run and vertical rise. Watch the double negative: 4 − (−2) = 4 + 2 = 6 km.
What happens: A right triangle with legs 6 km and 8 km.
2
$d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ km}$
Why: Use Pythagoras’ Theorem on the 6 km run and 8 km rise.
What happens: The straight-line flight is 10 km.
3
$M = \left(\frac{-2 + 4}{2},\; \frac{1 + 9}{2}\right) = (1,\; 5)$
Why: Midpoint is the halfway average: ADD the x-coordinates and divide by 2, then ADD the y-coordinates and divide by 2.
What happens: The halfway relay sits at (1, 5).
4
$\text{Trap: } \Delta x = 4 - 2 = 2 \text{ km}\,?$
Why: The glitch: dropping the minus sign on −2. From 2 km west (−2) to 4 km east (+4) is 6 km across the axis, not 2 km!
What happens: Always wrap negative coordinates in brackets: 4 − (−2) = 6 ✓.

Common glitch

Three classic coordinate-geometry glitches:

1.Putting Run ( $x$ ) on top of Rise ( $y$ ) in the gradient formula

Wrong: $m = \frac{x_2 - x_1}{y_2 - y_1}$ - upside down!

Right:Rise before you Run: vertical change $y_2 - y_1$ always goes on top of horizontal change $x_2 - x_1$ .

2.Subtracting instead of adding in the Midpoint formula

Wrong: $M = \left(\frac{x_2 - x_1}{2},\; \frac{y_2 - y_1}{2}\right)$ - that finds half the distance, not the midpoint

Right:Midpoint is the AVERAGE of the ends, so ADD the coordinates: $\left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)$ .

3.Dropping the minus sign when subtracting a negative coordinate

Wrong:Finding the run between $x = -2$ and $x = 4$ as $4 - 2 = 2$

Right:Use brackets around negative coordinates: $4 - (-2) = 4 + 2 = 6$ .

Linear & Quadratic Graphing Calculator - Plot straight lines y = mx + c and inspect their intercepts.
Pythagorean Theorem Calculator - Compute the straight-line distance between two points from Δx and Δy.

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Common glitch

Three classic coordinate-geometry glitches:

1.Putting Run ( $x$ ) on top of Rise ( $y$ ) in the gradient formula

Wrong: $m = \frac{x_2 - x_1}{y_2 - y_1}$ - upside down!

Right:Rise before you Run: vertical change $y_2 - y_1$ always goes on top of horizontal change $x_2 - x_1$ .

2.Subtracting instead of adding in the Midpoint formula

Wrong: $M = \left(\frac{x_2 - x_1}{2},\; \frac{y_2 - y_1}{2}\right)$ - that finds half the distance, not the midpoint

Right:Midpoint is the AVERAGE of the ends, so ADD the coordinates: $\left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)$ .

3.Dropping the minus sign when subtracting a negative coordinate

Wrong:Finding the run between $x = -2$ and $x = 4$ as $4 - 2 = 2$

Right:Use brackets around negative coordinates: $4 - (-2) = 4 + 2 = 6$ .

Learn: Coordinate Geometry

The idea

The four coordinate tools:

Quick memory tips

Worked example

Real-world example

Common glitch

Try a related calculator

Learn: Coordinate Geometry

The idea

The four coordinate tools:

Quick memory tips

Worked example

Real-world example

Common glitch

Try a related calculator