The Distance Formula

What is the Distance Formula?

The distance formula calculates the straight-line distance between two points on a coordinate plane.

Formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Where:

• (x₁, y₁) = first point
• (x₂, y₂) = second point
• d = distance

Comes from: Pythagorean Theorem!

How It Works

Think of the distance as the hypotenuse of a right triangle:

• Horizontal leg: |x₂ − x₁|
• Vertical leg: |y₂ − y₁|
• Hypotenuse: Distance between points

Pythagorean Theorem: a² + b² = c²

Becomes: (x₂ − x₁)² + (y₂ − y₁)² = d²

Solve for d: d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Using the Distance Formula

Example 1: Basic Distance

Find distance between (1, 2) and (4, 6)

Step 1: Identify coordinates

• (x₁, y₁) = (1, 2)
• (x₂, y₂) = (4, 6)

Step 2: Substitute into formula

d = √[(4 − 1)² + (6 − 2)²]
d = √[3² + 4²]
d = √[9 + 16]
d = √25
d = 5

Answer: 5 units

Example 2: Points in Different Quadrants

Find distance between (−3, 4) and (2, −8)

Step 1: Identify coordinates

• (x₁, y₁) = (−3, 4)
• (x₂, y₂) = (2, −8)

Step 2: Calculate

d = √[(2 − (−3))² + (−8 − 4)²]
d = √[(2 + 3)² + (−12)²]
d = √[5² + 144]
d = √[25 + 144]
d = √169
d = 13

Answer: 13 units

Example 3: Decimal Answer

Find distance between (1, 1) and (4, 5)

Calculate:

d = √[(4 − 1)² + (5 − 1)²]
d = √[3² + 4²]
d = √[9 + 16]
d = √25
d = 5

Answer: 5 units

Example 4: Non-Perfect Square

Find distance between (0, 0) and (3, 4)

Calculate:

d = √[(3 − 0)² + (4 − 0)²]
d = √[9 + 16]
d = √25
d = 5

Answer: 5 units (This is a 3-4-5 right triangle!)

Example 5: Approximate Answer

Find distance between (2, 3) and (5, 7)

Calculate:

d = √[(5 − 2)² + (7 − 3)²]
d = √[3² + 4²]
d = √[9 + 16]
d = √25
d = 5

Answer: 5 units

Special Cases

Horizontal Distance

Points have same y-coordinate

Example: (1, 3) and (7, 3)

Distance: |7 − 1| = 6 units

Using formula:

d = √[(7 − 1)² + (3 − 3)²]
d = √[36 + 0]
d = 6

Shortcut: Just subtract x-coordinates!

Vertical Distance

Points have same x-coordinate

Example: (4, 2) and (4, 9)

Distance: |9 − 2| = 7 units

Using formula:

d = √[(4 − 4)² + (9 − 2)²]
d = √[0 + 49]
d = 7

Shortcut: Just subtract y-coordinates!

Order Doesn't Matter

You can use either point as (x₁, y₁)

Example: Same Result Both Ways

Points: A(2, 5) and B(6, 8)

Method 1: A as (x₁, y₁)

d = √[(6 − 2)² + (8 − 5)²]
d = √[16 + 9]
d = √25 = 5

Method 2: B as (x₁, y₁)

d = √[(2 − 6)² + (5 − 8)²]
d = √[(−4)² + (−3)²]
d = √[16 + 9]
d = √25 = 5

Same answer! Squaring eliminates the negative signs.

Using Pythagorean Theorem

Alternative method: Draw the triangle!

Example: Visual Method

Points: (1, 2) and (5, 5)

Step 1: Draw horizontal leg

• From (1, 2) to (5, 2)
• Length: 5 − 1 = 4

Step 2: Draw vertical leg

• From (5, 2) to (5, 5)
• Length: 5 − 2 = 3

Step 3: Use Pythagorean Theorem

d² = 4² + 3²
d² = 16 + 9
d² = 25
d = 5

Answer: 5 units

Finding Missing Coordinate

If you know distance and one point, find the missing coordinate!

Example: Find Missing y-coordinate

Point A: (1, 3) Point B: (4, y) Distance: 5

Set up equation:

5 = √[(4 − 1)² + (y − 3)²]
5 = √[9 + (y − 3)²]
25 = 9 + (y − 3)²
16 = (y − 3)²
±4 = y − 3

Two solutions:

• y = 3 + 4 = 7
• y = 3 − 4 = −1

Answer: y = 7 or y = −1 (two possible points)

Real-World Applications

GPS: Calculate distance between locations

• Coordinates on map → distance in miles/km

Video games: Distance between objects

• Character to enemy, object to target

Delivery routes: Find shortest paths

• Warehouse to customer locations

Sports: Track player movement

• Distance covered on field

Architecture: Measure diagonal distances

• Room layouts, building plans

Practice