Distance Formula

📏 The Distance Formula

Part 1 of 5 — Concept Introduction

How far apart are two points on a coordinate plane? You could try to measure with a ruler, but that is slow and never exact. The distance formula gives you the precise straight-line distance between any two points, every single time.

The best part? You already know where it comes from — the Pythagorean Theorem!

The Formula

Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ between them is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

In plain English: subtract the x's, subtract the y's, square each difference, add them up, then take the square root.

Where Does It Come From? 🔺

Picture two points on a graph. Connect them with a straight line — that line is the hypotenuse of a right triangle. The two legs run perfectly horizontal and vertical.

Part of the triangle	What it represents	Length
Horizontal leg	Difference in x-values	$\|x_2 - x_1\|$
Vertical leg	Difference in y-values	$\|y_2 - y_1\|$
Hypotenuse	The distance we want	$d$

The Pythagorean Theorem says $\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2$, so:

$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$

Take the square root of both sides and you get the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

💡 Why squaring saves us: Even if a difference comes out negative, squaring it makes it positive — so the order you subtract in does not change the answer.

A First Worked Example

Find the distance between $(1, 2)$ and $(4, 6)$.

Step 1 — Label the points. $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 6)$.

Step 2 — Plug into the formula. $d = \sqrt{(4 - 1)^2 + (6 - 2)^2}$

Step 3 — Simplify inside. $d = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16}$

Step 4 — Add, then take the square root. $d = \sqrt{25} = 5$

Answer: $5$ units. Notice this is the famous $3$-$4$-$5$ right triangle hiding in the coordinate plane!

Concept Check 🎯

Make sure you understand what the distance formula is built from.

📝 Worked Examples

Part 2 of 5 — Step by Step

The same four steps work every time. Watch what happens when the numbers are not so friendly.

Example: Distance Between $(-2, 3)$ and $(4, -1)$

Step 1 — Label. $(x_1, y_1) = (-2, 3)$ and $(x_2, y_2) = (4, -1)$.

Step 2 — Substitute. Be careful subtracting negatives! $d = \sqrt{(4 - (-2))^2 + (-1 - 3)^2}$

Step 3 — Simplify the subtractions. $d = \sqrt{(4 + 2)^2 + (-4)^2} = \sqrt{(6)^2 + (-4)^2}$

Step 4 — Square, add, and take the root. $d = \sqrt{36 + 16} = \sqrt{52}$

Step 5 — Simplify the radical. Since $52 = 4 \times 13$: $d = \sqrt{4 \times 13} = 2\sqrt{13} \approx 7.21$

Answer: $2\sqrt{13}$ units (about $7.21$ units).

💡 Subtracting a negative is the same as adding: $4 - (-2) = 4 + 2 = 6$.

Your Turn 🧮

Find the distance between $(0, 0)$ and $(6, 8)$.

1. What is the horizontal difference $(x_2 - x_1)$? (Type a single number.)

2. What is the vertical difference $(y_2 - y_1)$? (Type a single number.)

3. What is the final distance $d = \sqrt{6^2 + 8^2}$? (Type a single number.)

🧭 Guided Practice

Part 3 of 5 — Guided Practice

Work each one carefully. Watch the signs and remember to square before you add.

Fill In the Steps 🔍

Complete the work for finding the distance between $(1, 3)$ and $(7, 3)$.

🗺️ Real-World Applications

Part 4 of 5 — Word Problems

The distance formula is not just for math class — it powers maps, video games, and GPS systems!

City Map Example

Mia's town is laid out on a grid where each unit is one city block. Her house is at $(2, 1)$ and the library is at $(5, 5)$. How many blocks is the straight-line ("as the crow flies") distance?

Set it up: $d = \sqrt{(5 - 2)^2 + (5 - 1)^2}$

Simplify: $d = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

So the library is $5$ blocks away in a straight line. A bird could fly there in 5 blocks, even though Mia might have to walk farther along the streets!

Treasure Hunt 🏴‍☠️

On a treasure map, the palm tree is at $(-1, -2)$ and the buried chest is at $(2, 2)$. Each unit is one step.

1. What is the horizontal difference $(x_2 - x_1) = 2 - (-1)$? (Type a single number.)

2. What is the vertical difference $(y_2 - y_1) = 2 - (-2)$? (Type a single number.)

3. How many steps is the chest from the tree, $d = \sqrt{3^2 + 4^2}$? (Type a single number.)

Application Check 🎮

In a video game, a player is at point $(0, 0)$ and a coin is at $(8, 6)$.

🏆 Review & Challenge

Part 5 of 5 — Putting It All Together

You have mastered the distance formula! Here is a quick reference summary.

Summary Table

Situation	Shortcut	Example
General case	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	$(1,2)$ to $(4,6)$ → $5$
Same y (horizontal)	$d = \|x_2 - x_1\|$	$(1,3)$ to $(7,3)$ → $6$
Same x (vertical)	$d = \|y_2 - y_1\|$	$(2,1)$ to $(2,8)$ → $7$
From the origin	$d = \sqrt{x^2 + y^2}$	$(0,0)$ to $(5,12)$ → $13$

Key Reminders

• Subtract, square, add, root — in that order.
• Squaring erases negatives, so the order of subtraction does not matter.
• Watch out for subtracting negatives: $a - (-b) = a + b$.
• Look for special triangles like $3$-$4$-$5$ and $5$-$12$-$13$ to check your work fast.

Challenge Round 🚀

Mix everything together. Take your time on the signs.