x₂ − x₁
This measures how far apart the two x-coordinates are horizontally.
Free Coordinate Geometry Guide
Distance Formula
Learn how to label two coordinate points, substitute into the distance formula, simplify negative-number expressions, evaluate square roots, and understand why the formula comes from the Pythagorean theorem.
The distance formula finds the straight-line distance between two points on the coordinate plane. It combines horizontal and vertical coordinate changes into one distance.
The formula may look complicated at first, but it becomes much easier when you label the coordinates first and substitute one part at a time.
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What Is the Distance Formula?
If two points are written as (x₁, y₁) and (x₂, y₂), the distance between them is:
d =
√
(x₂ − x₁)2
+
(y₂ − y₁)2
The formula takes the difference between the x-coordinates, squares it, takes the difference between the y-coordinates, squares that, adds the results, and then takes the square root.
y₂ − y₁
This measures how far apart the two y-coordinates are vertically.
Label x₁, y₁, x₂, and y₂ First
Before substituting anything into the formula, label the coordinates. This is one of the easiest ways to prevent mixing up x- and y-values.
(x₁, y₁) = (2, 3)
(x₂, y₂) = (8, 11)
| Variable | Meaning | Value |
|---|---|---|
| x₁ | x-coordinate of Point 1 | 2 |
| y₁ | y-coordinate of Point 1 | 3 |
| x₂ | x-coordinate of Point 2 | 8 |
| y₂ | y-coordinate of Point 2 | 11 |
A useful habit:
keep the first point together and keep the
second point together.
How to Use the Distance Formula
Suppose we want the distance between (2, 3) and (8, 11).
1. Label the points
(x₁, y₁) = (2, 3) and
(x₂, y₂) = (8, 11)
2. Substitute x-values
(x₂ − x₁)² becomes
(8 − 2)².
3. Substitute y-values
(y₂ − y₁)² becomes
(11 − 3)².
4. Simplify differences
8 − 2 = 6 and 11 − 3 = 8.
5. Square and add
6² + 8² = 36 + 64 = 100.
6. Take the square root
√
100
= 10.
d =
√
(8 − 2)2
+
(11 − 3)2
d =
√
62
+
82
d =
√
100
d = 10
The Distance Formula Comes From the Pythagorean Theorem
The distance formula is not a completely separate idea. It is the Pythagorean theorem applied to the coordinate plane.
a² + b² = c²
The difference between the x-coordinates creates the horizontal leg of a right triangle. The difference between the y-coordinates creates the vertical leg. The segment connecting the two original points is the hypotenuse.
Distance as a Right Triangle
The points (2, 3) and (8, 11) create horizontal and vertical changes of 6 and 8.
The horizontal leg has length 6, the vertical leg has length 8, and the hypotenuse has length 10.
The key connection:
The distance formula is simply the Pythagorean theorem with x₂ − x₁ and y₂ − y₁ used for the two legs.
Distance Formula With Negative Coordinates
Negative coordinates do not change the formula. The challenge is substituting them correctly.
Suppose the points are (−3, 4) and (5, −2).
(x₁, y₁) = (−3, 4)
(x₂, y₂) = (5, −2)
(x₂, y₂) = (5, −2)
Substitute carefully:
d =
√
(5 − (−3))2
+
(−2 − 4)2
Notice the first difference:
5 − (−3) = 8
Subtracting a negative becomes addition. The second difference is:
−2 − 4 = −6
The negative difference is not a problem because it will be squared:
d =
√
82
+
(−6)2
d =
√
64
+
36
d =
√
100
d = 10
Use parentheses around negative values.
Writing 5 − (−3) makes the substitution much clearer than trying to handle multiple signs mentally.
Square Roots and Exact Distance Answers
Distance problems do not always produce whole numbers. Sometimes the final answer is a square root.
√
100
= 10
If the number under the radical is a perfect square, simplify to a whole number.
√
13
If no perfect-square factor greater than 1 divides the radicand, the radical remains unchanged.
√
20
= 2
√
5
Factor out the largest useful perfect square and simplify.
√
13
≈ 3.61
A decimal may be useful, but the exact radical form is usually preferred unless the problem asks for an approximation.
Exact form first.
If the distance is irrational, keep the exact radical answer unless directions specifically request a decimal.
Does It Matter Which Point Is Point 1?
No. Either point can be labeled Point 1, as long as you label the other point consistently as Point 2.
For example:
8 − 2 = 6
2 − 8 = −6
2 − 8 = −6
The signs are different, but squaring produces the same result:
6² = 36
(−6)² = 36
(−6)² = 36
The important part is consistency.
Do not use x₂ − x₁ for one coordinate pair and then switch to y₁ − y₂ unintentionally. A consistent point order keeps the work easier to follow.
Worked Distance Formula Examples
Find the distance from (1, 2) to (4, 6)
- Label: (x₁, y₁) = (1, 2), (x₂, y₂) = (4, 6).
- Substitute: √ (4 − 1)2 + (6 − 2)2 .
- Simplify: √ 32 + 42 .
- √ 9 + 16 = √ 25 .
- Answer: 5 units.
Find the distance from (−2, 1) to (4, −7)
- Label both coordinate pairs.
- Substitute: √ (4 − (−2))2 + (−7 − 1)2 .
- Simplify: √ 62 + (−8)2 .
- √ 36 + 64 = √ 100 .
- Answer: 10 units.
Find the distance from (0, 0) to (2, 3)
- Substitute: √ (2 − 0)2 + (3 − 0)2 .
- Simplify: √ 22 + 32 .
- √ 4 + 9 = √ 13 .
- 13 has no perfect-square factor greater than 1.
- Answer: √ 13 units .
Find the distance from (1, 1) to (3, 5)
- Substitute: √ (3 − 1)2 + (5 − 1)2 .
- Simplify: √ 22 + 42 .
- √ 4 + 16 = √ 20 .
- √ 20 = √ 4 × 5 = 2 √ 5 .
- Answer: 2 √ 5 units .
Common Distance Formula Mistakes
Mistake 1: Mixing x- and y-coordinates
The first difference must use the two x-coordinates. The second difference must use the two y-coordinates.
Mistake 2: Forgetting parentheses around negatives
If x₁ = −3, then x₂ − x₁ should be written as x₂ − (−3). Parentheses make the sign operation clear.
Mistake 3: Squaring before subtracting
In (x₂ − x₁)², simplify the subtraction inside the parentheses first, then square.
Mistake 4: Forgetting to square both differences
Both the x-difference and the y-difference are squared before they are added.
Mistake 5: Forgetting the final square root
The sum of the squares is not the final distance. The last step is taking the square root.
Mistake 6: Giving an unsimplified radical
If the result is √ 20 , simplify it to 2 √ 5 unless the directions say otherwise.
Mistake 7: Rounding too early
Keep the exact radical through the calculation and round only at the end if a decimal approximation is requested.
A Reliable Distance Formula Strategy
1. Label Point 1
Write the first point as
(x₁, y₁).
2. Label Point 2
Write the second point as
(x₂, y₂).
3. Substitute the x-values
Fill in
x₂ − x₁.
4. Substitute the y-values
Fill in
y₂ − y₁.
5. Simplify the differences
Perform the subtraction inside each pair
of parentheses.
6. Square and add
Square both differences and add the
results.
7. Take the square root
Simplify the radical to give the exact
distance.
The safest habit:
label first, substitute second, simplify last.
Practice What You Learned
Try the Distance Formula Interactive Tool
Practice labeling x₁, y₁, x₂, and y₂, substituting each value into the formula, simplifying coordinate differences, handling negative coordinates, and finding exact distance answers step by step.
Start Distance Formula PracticeRelated Coordinate Geometry Resources
The distance formula connects coordinate-plane skills with slope, graphing, and other coordinate geometry topics.
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