EQN-04-06

Coordinate Geometry

Distance, midpoint, and slope on the xy-plane — three formulas that answer most coordinate questions.

~6 min read•Difficulty 2/5

The three formulas

For points (x₁, y₁) and (x₂, y₂):

distance = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} (Pythagoras in disguise)

midpoint = ((x_{1} + x_{2}) /2, (y_{1} + y_{2}) /2) (average the coordinates)

slope = (y_{2} - y_{1}) / (x_{2} - x_{1}) (rise over run)

What slope tells you

Positive slope rises left-to-right; negative falls.
Slope 0 is horizontal; a vertical line has an undefined slope.
Parallel lines share the same slope; perpendicular slopes multiply to $- 1.$

Line equation

y = mx + b (m = slope, b = y -intercept where the line crosses the y -axis)

Read the signs carefully

The distance and midpoint formulas survive negative coordinates because of the subtraction and squaring — but a sign slip in $(x_{2} - x_{1})$ is the usual error. Keep the order consistent.

Diagram

Distance comes from the right triangle formed by $Δ x$ and $Δ y .$

Worked examples

Example 1 (difficulty 2/5)

What is the distance between (1, 2) and (4, 6)?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

What is the slope of the line through

(- 2, 1)

and (2, 9)?

◎ Predict the answer first:

Tips

The distance formula is just Pythagoras on $Δ x$ and $Δ y$ — look for 3-4-5 and 5-12-13.
Midpoint $=$ average of the coordinates; distance/slope use differences.
Parallel $\to$ equal slopes; perpendicular $\to$ slopes multiply to $- 1.$

Common mistakes

Mixing the order of subtraction in the slope (using x on top)

Why: Rise and run get swapped.

Fix: Slope is

(y_{2} - y_{1}) / (x_{2} - x_{1}) : y

-difference on top.

Dropping a negative sign with coordinates like

- 2

Why: Subtracting a negative is error-prone.

Fix:

2 - (- 2) = 4

; write the subtraction explicitly before simplifying.

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