Foundations — Distance formula — derivation using Pythagoras

Before we can even read the parent note Distance Formula, we need every single symbol it uses to feel obvious. Below, each idea is built from nothing, anchored to a picture, and shown to be needed. Read top to bottom — each block stands on the one above it.

1. The plane: a flat sheet of numbered space

The picture: think of the floor of a room. Pick one corner as your "start". "How far along the wall" is one number; "how far out from the wall" is a second number. Those two lines along the walls are your axes.

Why the topic needs it: the distance formula measures the gap between two locations. Before we can talk about a location, we need a fixed stage to place it on. That stage is the plane. See Cartesian Coordinate System for the full setup.

2. A coordinate: two numbers that pin a point

The picture: to reach the point $(3,2)$, start at the origin, walk 3 steps right, then 2 steps up. You've arrived. The order matters: $(3,2)$ is not the same spot as $(2,3)$ — that's why we call it an ordered pair.

Why the topic needs it: the parent note writes points as $P(x_1,y_1)$ and $Q(x_2,y_2)$. Without the idea that two numbers lock a location, those letters are meaningless.

3. Subscripts: telling two points apart

• $x_1,y_1$ = the two coordinates of the first point $P$.
• $x_2,y_2$ = the two coordinates of the second point $Q$.

The picture: imagine two houses. House 1 has an address $(x_1,y_1)$; House 2 has an address $(x_2,y_2)$. The "1" and "2" are just name tags on the doors.

Why the topic needs it: every difference in the formula, like $(x_2-x_1)$, relies on knowing which point each coordinate belongs to.

4. Difference of coordinates: horizontal and vertical gaps

The picture: stand at $P$ and look at $Q$. First measure only sideways movement (ignore up/down) — that's $x_2-x_1$. Then measure only up/down movement — that's $y_2-y_1$.

Why the topic needs it: these two differences become the two legs of the right triangle. They are the raw materials of the whole formula.

5. Sign and absolute value: distance is never negative

The picture: whether you walk 5 steps right or 5 steps left, you walked 5 steps. Direction (the sign) can be negative; length walked cannot.

Why the topic needs it: a horizontal gap $x_2-x_1$ might come out negative (if $Q$ is left of $P$), but a length must be positive. So the leg length is written $|x_2-x_1|$. Later, squaring will make this automatic — because a squared number is never negative.

6. Squares and square roots: the two undo-buttons

• $3^2=9$, and $\sqrt{9}=3$.
• Crucially, $(-3{)}^2=9$ too — squaring destroys the minus sign.

Why the topic needs it: Pythagoras works with squares of lengths, and the distance sits under a square root. Without both undo-buttons, the final formula is unreadable.

7. The right triangle and Pythagoras: the engine

The picture: the horizontal gap and the vertical gap between $P$ and $Q$ meet at a perfect right angle (because the $x$-axis and $y$-axis are perpendicular). The straight line from $P$ to $Q$ closes the triangle — and that closing line is the distance we want.

Why the topic needs it: this is the single fact that turns two coordinate gaps into one distance. Everything else on this page exists to make Pythagoras usable. Deep dive: Pythagoras Theorem.

8. Putting the symbols together

Now every symbol in the parent's formula has a home:

$d=\sqrt{\underset{\text{horizontal leg squared}}{\underbrace{(x_2-x_1{)}^2}}+\underset{\text{vertical leg squared}}{\underbrace{(y_2-y_1{)}^2}}}$

• $d$ = the distance = the hypotenuse.
• $(x_2-x_1)$ and $(y_2-y_1)$ = the two legs (differences).
• Squaring = apply Pythagoras and kill sign worries.
• ​ = undo the squaring to get back a length.

Prerequisite map

Each arrow means "you must understand this before the next makes sense". The two feeds into Pythagoras (from differences and from squares) are exactly the two legs and their squares.

Equipment checklist

Test yourself — say the answer out loud before revealing.

Connections

• Distance formula — derivation using Pythagoras — the parent this page prepares you for
• Cartesian Coordinate System — where points and axes are defined
• Pythagoras Theorem — the engine that combines the two legs
• Midpoint Formula — next step, also built on coordinate differences
• Section Formula — divides a segment using these same coordinates
• Equation of Circle — all points at a fixed distance from a centre
• Distance Formula in3D — adds a third coordinate difference