Distance on the Coordinate Plane

A coordinate plane allows us to describe the position of points using ordered pairs.

A point is written:

(x, y)

where:

• x gives the horizontal position
• y gives the vertical position

For example:

A(2, 3)

means that point A is:

• 2 units horizontally from the origin
• 3 units vertically from the origin

When two points are given, we can calculate the distance between them.

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Horizontal Distance

If two points have the same y-coordinate, the line between them is horizontal.

For example:

A(2, 4)

and:

B(8, 4)

The horizontal distance is:

8 − 2 = 6

Therefore:

AB = 6 units

In general, the horizontal distance between two points is:

|x₂ − x₁|

The absolute value ensures that distance is positive.

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Vertical Distance

If two points have the same x-coordinate, the line between them is vertical.

For example:

A(3, 2)

and:

B(3, 9)

The vertical distance is:

9 − 2 = 7

Therefore:

AB = 7 units

In general:

vertical distance = |y₂ − y₁|

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Why We Use Absolute Value

Suppose the points are:

A(8, 4)

and:

B(2, 4)

If we calculate:

2 − 8 = −6

But a distance cannot be negative.

Therefore:

|2 − 8| = 6

Distance represents a magnitude, so it is always zero or positive.

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When the Points Are Diagonal

Now consider:

A(1, 2)

and:

B(5, 5)

The points do not share the same x-coordinate or y-coordinate.

We cannot find the direct distance by simply subtracting one pair of coordinates.

Instead, we can construct a right triangle.

The horizontal and vertical changes become the two shorter sides of the triangle.

The direct distance between the points becomes the hypotenuse.

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Finding the Horizontal Change

For:

A(1, 2)

and:

B(5, 5)

the horizontal change is:

Δx = 5 − 1

Δx = 4

The symbol:

Δ

means change in.

Therefore:

Δx

means:

change in x

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Finding the Vertical Change

For the same points:

A(1, 2)

and:

B(5, 5)

the vertical change is:

Δy = 5 − 2

Δy = 3

We now have a right triangle with side lengths:

4

and:

3

The direct distance between A and B is the hypotenuse.

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The Pythagorean Theorem

The Pythagorean Theorem relates the three sides of a right triangle.

For a right triangle with shorter sides a and b, and hypotenuse c:

a² + b² = c²

In coordinate geometry:

• horizontal change acts as one shorter side
• vertical change acts as the other shorter side
• distance between the points acts as the hypotenuse

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Applying Pythagoras to Coordinates

Using:

A(1, 2)

and:

B(5, 5)

we found:

horizontal distance = 4

vertical distance = 3

Use Pythagoras:

d² = 4² + 3²

d² = 16 + 9

d² = 25

Take the square root:

d = √25

d = 5

Therefore:

AB = 5 units

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From Pythagoras to the Distance Formula

Suppose the two points are:

(x₁, y₁)

and:

(x₂, y₂)

The horizontal change is:

x₂ − x₁

The vertical change is:

y₂ − y₁

Using the Pythagorean Theorem:

d² = (x₂ − x₁)² + (y₂ − y₁)²

Taking the square root gives the distance formula:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

The distance formula is therefore not a completely separate idea.

It comes directly from the:

Pythagorean Theorem

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Understanding the Distance Formula

The formula is:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

where:

(x₁, y₁) = first point

(x₂, y₂) = second point

x₂ − x₁ = horizontal change

y₂ − y₁ = vertical change

d = straight-line distance between the points

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A Reliable Method

When finding the distance between two points:

Step 1: Label the coordinates.

Step 2: Find the horizontal difference.

Step 3: Find the vertical difference.

Step 4: Square both differences.

Step 5: Add the squares.

Step 6: Take the square root.

Step 7: Include appropriate units if required.

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Worked Example 1: Horizontal Distance

Find the distance between:

A(−3, 5)

and:

B(7, 5)

The y-coordinates are the same.

Therefore, the distance is horizontal.

d = |7 − (−3)|

d = |10|

d = 10

Answer:

10 units

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Worked Example 2: Vertical Distance

Find the distance between:

A(4, −2)

and:

B(4, 6)

The x-coordinates are the same.

Therefore:

d = |6 − (−2)|

d = 8

Answer:

8 units

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Worked Example 3: A 3-4-5 Triangle

Find the distance between:

A(2, 1)

and:

B(6, 4)

Horizontal change:

6 − 2 = 4

Vertical change:

4 − 1 = 3

Apply Pythagoras:

d² = 4² + 3²

d² = 16 + 9

d² = 25

Therefore:

d = 5

Answer:

5 units

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Worked Example 4: Using the Distance Formula

Find the distance between:

A(1, 3)

and:

B(7, 11)

Use:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Substitute:

d = √[(7 − 1)² + (11 − 3)²]

Simplify:

d = √[6² + 8²]

d = √[36 + 64]

d = √100

d = 10

Answer:

10 units

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Negative Coordinates

The distance formula works exactly the same way when coordinates are negative.

Suppose:

A(−4, 2)

and:

B(3, 2)

Horizontal difference:

3 − (−4)

Remember:

Subtracting a negative becomes addition.

Therefore:

3 + 4 = 7

Distance:

7 units

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Worked Example 5: Negative Coordinates

Find the distance between:

A(−2, −1)

and:

B(4, 7)

Horizontal change:

4 − (−2) = 6

Vertical change:

7 − (−1) = 8

Distance:

d = √(6² + 8²)

d = √(36 + 64)

d = √100

d = 10

Answer:

10 units

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Why Squaring Helps

Suppose:

x₂ − x₁ = −5

When squared:

(−5)² = 25

Suppose instead:

x₂ − x₁ = 5

Then:

5² = 25

Therefore, the distance formula gives the same answer regardless of which point is labeled first.

This makes sense because:

distance from A to B = distance from B to A

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Exact and Approximate Distances

Not every distance produces a whole number.

Suppose:

A(1, 1)

and:

B(4, 5)

Then:

d = √[(4 − 1)² + (5 − 1)²]

d = √(3² + 4²)

d = 5

But consider:

A(1, 1)

and:

B(4, 4)

Then:

d = √(3² + 3²)

d = √18

This can be simplified:

d = 3√2

or approximated:

d ≈ 4.24

units.

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Exact Form

An answer such as:

√13

is called an exact value.

It has not been rounded.

For example:

d = √13 units

may be the preferred answer when exact form is requested.

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Approximate Form

Using a calculator:

√13 ≈ 3.61

Therefore:

d ≈ 3.61 units

is an approximate value.

The symbol:

≈

means:

approximately equal to

Do not use = when giving a rounded approximation.

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Worked Example 6: An Irrational Distance

Find the distance between:

A(2, 3)

and:

B(7, 5)

Horizontal change:

7 − 2 = 5

Vertical change:

5 − 3 = 2

Distance:

d = √(5² + 2²)

d = √(25 + 4)

d = √29

Approximate value:

d ≈ 5.39

Answer:

√29 units

or approximately:

5.39 units

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Visualizing the Right Triangle

Whenever two points are diagonal from one another, imagine drawing:

1. a horizontal line from one point
2. a vertical line toward the other point
3. the direct line joining the original points

This creates a right triangle.

The horizontal and vertical changes are the legs.

The direct distance is the hypotenuse.

This picture explains the entire distance formula.

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Horizontal and Vertical Change

A useful notation is:

Δx = x₂ − x₁

and:

Δy = y₂ − y₁

Then the distance formula can be understood as:

d = √[(Δx)² + (Δy)²]

This emphasizes that distance depends on the horizontal and vertical changes between the points.

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Distance Is Not the Same as Horizontal Change

Suppose two points differ by:

6 units horizontally

and:

8 units vertically

The direct distance is not:

6 + 8 = 14

Instead:

d = √(6² + 8²)

d = √100

d = 10

The straight-line distance is:

10 units

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Straight-Line Distance vs Travel Distance

This distinction becomes important in real-world situations.

Suppose a person travels:

6 km east

and then:

8 km north

Total distance travelled:

6 + 8 = 14 km

But the person's straight-line distance from the starting point is:

10 km

These are different quantities.

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Distance and the Coordinate Plane

The coordinate plane can represent many real situations.

Coordinates might represent:

• locations on a map
• positions of buildings
• positions of objects
• locations in a game
• points on a sports field
• locations in an engineering design
• positions measured by sensors

The distance formula can then be used to calculate the direct distance between two locations.

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Real-World Example: A Park Map

Suppose a map uses coordinates measured in kilometres.

A visitor centre is located at:

V(2, 3)

A campsite is located at:

C(8, 11)

Horizontal difference:

8 − 2 = 6 km

Vertical difference:

11 − 3 = 8 km

Straight-line distance:

d = √(6² + 8²)

d = 10 km

Therefore, the visitor centre and campsite are:

10 km apart in a straight line

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Real-World Example: City Blocks

Suppose two buildings are located at:

A(1, 2)

and:

B(7, 10)

If each coordinate unit represents:

100 m

then:

Horizontal change:

6 units = 600 m

Vertical change:

8 units = 800 m

Straight-line coordinate distance:

10 units

Therefore:

10 × 100 = 1000 m

The buildings are:

1 km apart in a straight line

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Scale Matters

A coordinate answer of:

5 units

does not always mean:

5 metres

The meaning depends on the scale.

For example:

1 coordinate unit = 2 km

Then:

5 coordinate units = 10 km

Always check what each coordinate unit represents.

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Real-World Example: Computer Graphics

Computer screens and digital images use coordinate systems to identify locations.

If two objects are located at different coordinates, software can calculate the distance between them.

This can be useful for:

• collision detection
• animation
• games
• mapping
• robotics
• image processing

The same mathematical idea is being used:

horizontal difference + vertical difference → straight-line distance

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Real-World Example: Robotics

Imagine a robot is at:

(2, 4)

and needs to move directly to:

(8, 12)

Horizontal change:

6

Vertical change:

8

Direct distance:

d = √(6² + 8²)

d = 10

If each unit represents one metre:

the robot must travel 10 m

assuming it can move directly between the points.

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Real-World Example: Sports

A sports field can also be represented using coordinates.

Suppose two players are at:

P(5, 4)

and:

Q(17, 9)

Horizontal difference:

12

Vertical difference:

5

Distance:

d = √(12² + 5²)

d = √169

d = 13

If coordinates are measured in metres:

the players are 13 m apart.

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Worked Example 7: Across Quadrants

Find the distance between:

A(−5, 3)

and:

B(4, −9)

Horizontal change:

4 − (−5) = 9

Vertical change:

−9 − 3 = −12

Square the changes:

9² = 81

(−12)² = 144

Add:

81 + 144 = 225

Square root:

√225 = 15

Answer:

15 units

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Worked Example 8: Decimal Coordinates

Find the distance between:

A(1.5, 2)

and:

B(5.5, 5)

Horizontal change:

5.5 − 1.5 = 4

Vertical change:

5 − 2 = 3

Distance:

d = √(4² + 3²)

d = 5

Answer:

5 units

The distance formula works with decimal coordinates as well.

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Worked Example 9: Finding a Missing Distance

A triangle on a coordinate grid has a horizontal change of:

9 units

and a vertical change of:

12 units

Find the direct distance.

d = √(9² + 12²)

d = √(81 + 144)

d = √225

d = 15

Answer:

15 units

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Worked Example 10: Real-World Scale

Two points on a map are:

A(−1, 2)

and:

B(5, 10)

Each coordinate unit represents:

250 m

Coordinate distance:

d = √[(5 − (−1))² + (10 − 2)²]

d = √(6² + 8²)

d = 10 units

Convert:

10 × 250 = 2500 m

Therefore:

distance = 2500 m

or:

2.5 km

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Checking Whether an Answer Is Reasonable

Suppose the horizontal difference is:

4 units

and the vertical difference is:

7 units

The direct distance must be:

• greater than 7 units
• less than 11 units

Why?

It must be longer than either individual leg of the right triangle.

But it must be shorter than travelling the full horizontal and vertical path:

4 + 7 = 11

So an answer such as:

8.1 units

could be reasonable.

An answer such as:

3 units

could not be correct.

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Estimating Before Calculating

Consider:

Δx = 6

and:

Δy = 7

The distance must be slightly greater than:

7

but less than:

13

Calculate:

d = √(6² + 7²)

d = √85

d ≈ 9.22

The result fits our estimate.

Estimation is a useful way to detect calculator or substitution errors.

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Distance and Units

Always include units when the coordinate system represents a real measurement.

Examples:

8 m

14 km

5.6 cm

250 pixels

If the coordinate system has no specified measurement unit, write:

8 units

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Using a Calculator

For:

d = √[(8 − 2)² + (9 − 4)²]

calculate the differences first:

d = √(6² + 5²)

Then square:

d = √(36 + 25)

Add:

d = √61

Then calculate:

d ≈ 7.81

Keeping the work organized reduces calculator mistakes.

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Distance Formula and Pythagorean Theorem

The distance formula should not simply be memorized without understanding where it comes from.

The relationship is:

horizontal difference = one leg

vertical difference = second leg

distance = hypotenuse

Therefore:

Pythagorean Theorem → Distance Formula

Understanding this relationship makes the formula much easier to remember.

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A Problem-Solving Strategy

For coordinate-distance problems:

1. Identify both points.

2. Label x₁, y₁, x₂, and y₂.

3. Determine Δx.

4. Determine Δy.

5. Use Pythagoras or the distance formula.

6. Simplify the square root if possible.

7. Approximate if required.

8. Include appropriate units.

9. Interpret what the answer means in context.

10. Check whether the answer is reasonable.

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Common Mistakes

Mistake 1: Adding the horizontal and vertical distances

If the question asks for the direct straight-line distance, use Pythagoras.

Do not simply calculate:

Δx + Δy

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Mistake 2: Forgetting negative signs

For example:

5 − (−3)

equals:

8

not:

2

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Mistake 3: Mixing the coordinates

If you calculate:

x₂ − x₁

then compare x-coordinates.

If you calculate:

y₂ − y₁

then compare y-coordinates.

Do not subtract an x-coordinate from a y-coordinate.

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Mistake 4: Forgetting to square

Incorrect:

d = √[(x₂ − x₁) + (y₂ − y₁)]

Correct:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

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Mistake 5: Forgetting the square root

If:

d² = 100

then:

d = 10

not:

100

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Mistake 6: Reporting a negative distance

Distance cannot be negative.

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Mistake 7: Rounding too early

Keep exact values during intermediate calculations.

Round only at the end unless instructed otherwise.

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Mistake 8: Ignoring the scale

A distance of:

6 coordinate units

might represent:

6 m

600 m

12 km

or another value depending on the scale.

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Key Terms

• Coordinate plane: Two-dimensional system using perpendicular x- and y-axes.
• Ordered pair: Pair of coordinates written as (x, y).
• x-coordinate: Horizontal position of a point.
• y-coordinate: Vertical position of a point.
• Horizontal distance: Difference between x-coordinates.
• Vertical distance: Difference between y-coordinates.
• Distance: Length of the straight-line segment joining two points.
• Pythagorean Theorem: Relationship between the side lengths of a right triangle.
• Hypotenuse: Longest side of a right triangle, opposite the right angle.
• Δx: Change in the x-coordinate.
• Δy: Change in the y-coordinate.
• Distance formula: Formula used to calculate the straight-line distance between two coordinate points.
• Exact value: Value that has not been rounded, such as √13.
• Approximation: Rounded value close to the exact value.
• Scale: Relationship between coordinate units and real-world measurements.
• Magnitude: Size or amount of a quantity without direction.

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Key Formulas

Horizontal distance:

|x₂ − x₁|

Vertical distance:

|y₂ − y₁|

Pythagorean Theorem:

a² + b² = c²

Distance Formula:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

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Key Takeaways

• Coordinates describe positions using x and y values.
• Horizontal distance is found from the difference between the x-coordinates.
• Vertical distance is found from the difference between the y-coordinates.
• Absolute value can be used when calculating purely horizontal or vertical distances because distance cannot be negative.
• Two points that differ in both x and y can be connected using a right triangle.
• The horizontal and vertical differences form the two shorter sides of that triangle.
• The straight-line distance forms the hypotenuse.
• The Pythagorean Theorem is therefore the basis of the distance formula.
• The distance formula is d = √[(x₂ − x₁)² + (y₂ − y₁)²].
• Negative coordinates do not change the method.
• Squaring the coordinate differences ensures their signs do not produce a negative distance.
• Some distances are whole numbers, while others are irrational numbers that can be left in radical form or approximated.
• Distance from A to B is the same as distance from B to A.
• Straight-line distance is different from the total distance travelled along horizontal and vertical paths.
• Coordinate distance can represent real distances on maps, sports fields, computer screens, engineering plans, and other coordinate systems.
• The scale of a coordinate system must be considered when interpreting a distance.
• A strong solution should include the calculation, appropriate units, and an interpretation of what the distance means.