1. Distance Between Two Points

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.

---

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.

2. The Midpoint of a Line Segment

What Is a Midpoint?

A line segment is the part of a line between two endpoints.

The midpoint is the point exactly halfway between those endpoints.

If a line segment has endpoints A and B, and M is the midpoint, then:

AM = MB

This means that the midpoint divides the original segment into two equal-length segments.

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Midpoint Means Halfway

Suppose a line segment is:

10 cm long

Its midpoint is:

5 cm from either endpoint

If the entire segment is 18 m long, the midpoint is:

9 m from either endpoint

The same idea applies to points on a coordinate plane.

The midpoint must be halfway between the endpoints both:

horizontally

and:

vertically

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Midpoints on a Number Line

Before working in two dimensions, consider a number line.

Find the midpoint between:

2 and 8

Add the numbers:

2 + 8 = 10

Divide by 2:

10 ÷ 2 = 5

Therefore, the midpoint is:

5

This is simply the average of the two endpoints.

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Another Number-Line Example

Find the midpoint between:

−4 and 10

Add:

−4 + 10 = 6

Divide by 2:

6 ÷ 2 = 3

Therefore, the midpoint is:

3

Notice that negative numbers do not change the method.

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Midpoints on the Coordinate Plane

Now consider two points:

A(2, 4)

and:

B(8, 10)

The midpoint must be halfway between the x-coordinates and halfway between the y-coordinates.

Average the x-coordinates:

(2 + 8) ÷ 2 = 5

Average the y-coordinates:

(4 + 10) ÷ 2 = 7

Therefore:

M(5, 7)

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The Midpoint Formula

If the endpoints are:

A(x₁, y₁)

and:

B(x₂, y₂)

then the midpoint is:

M((x₁ + x₂)/2, (y₁ + y₂)/2)

In words:

average the x-coordinates

and:

average the y-coordinates

That is the entire idea behind the midpoint formula.

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Why Do We Average the Coordinates?

Suppose:

x₁ = 2

and:

x₂ = 8

The number halfway between them is:

5

because:

5 − 2 = 3

and:

8 − 5 = 3

Similarly, if:

y₁ = 4

and:

y₂ = 10

the halfway value is:

7

because:

7 − 4 = 3

and:

10 − 7 = 3

Therefore:

(5, 7)

is exactly halfway between:

(2, 4)

and:

(8, 10)

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A Reliable Midpoint Method

When finding a midpoint:

Step 1: Identify both endpoints.

Step 2: Add the x-coordinates.

Step 3: Divide the result by 2.

Step 4: Add the y-coordinates.

Step 5: Divide the result by 2.

Step 6: Write the answer as an ordered pair.

A useful reminder is:

Average x with x and y with y.

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Worked Example 1: Positive Coordinates

Find the midpoint of:

A(2, 6)

and:

B(10, 12)

Average x:

(2 + 10)/2 = 12/2 = 6

Average y:

(6 + 12)/2 = 18/2 = 9

Therefore:

M(6, 9)

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Worked Example 2: Another Example

Find the midpoint between:

A(1, 3)

and:

B(7, 11)

Average x:

(1 + 7)/2 = 4

Average y:

(3 + 11)/2 = 7

Therefore:

M(4, 7)

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Negative Coordinates

The midpoint formula works exactly the same way with negative coordinates.

Suppose:

A(−6, 2)

and:

B(4, 8)

Average x:

(−6 + 4)/2

= −2/2

= −1

Average y:

(2 + 8)/2

= 10/2

= 5

Therefore:

M(−1, 5)

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Worked Example 3: Both Coordinates Negative

Find the midpoint between:

A(−8, −4)

and:

B(−2, −10)

Average x:

(−8 + −2)/2

= −10/2

= −5

Average y:

(−4 + −10)/2

= −14/2

= −7

Therefore:

M(−5, −7)

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Midpoints Can Have Decimal Coordinates

The midpoint does not have to have whole-number coordinates.

Consider:

A(2, 3)

and:

B(7, 8)

Average x:

(2 + 7)/2 = 9/2 = 4.5

Average y:

(3 + 8)/2 = 11/2 = 5.5

Therefore:

M(4.5, 5.5)

This is perfectly valid.

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Horizontal Line Segments

Consider:

A(2, 5)

and:

B(10, 5)

The y-coordinates are the same.

Average x:

(2 + 10)/2 = 6

Average y:

(5 + 5)/2 = 5

Therefore:

M(6, 5)

The midpoint remains on the same horizontal line.

---

Vertical Line Segments

Consider:

A(4, −2)

and:

B(4, 10)

Average x:

(4 + 4)/2 = 4

Average y:

(−2 + 10)/2 = 4

Therefore:

M(4, 4)

The midpoint remains on the same vertical line.

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Checking a Midpoint

A midpoint should be the same distance from both endpoints.

Suppose:

A(1, 2)

B(7, 10)

and the calculated midpoint is:

M(4, 6)

From A to M:

Horizontal change:

4 − 1 = 3

Vertical change:

6 − 2 = 4

From M to B:

Horizontal change:

7 − 4 = 3

Vertical change:

10 − 6 = 4

Both halves have the same horizontal and vertical changes.

Therefore, M is the midpoint.

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Midpoint and Distance

The midpoint divides a segment into two equal lengths.

If:

AB = 14 units

and M is the midpoint, then:

AM = 7 units

and:

MB = 7 units

Therefore:

AM = MB = AB/2

This connects midpoint calculations with the distance formula.

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Finding a Missing Endpoint

Sometimes we know:

• one endpoint
• the midpoint

but need to determine the other endpoint.

Suppose:

A(2, 4)

Midpoint:

M(5, 7)

Find endpoint:

B(x, y)

We know:

(2 + x)/2 = 5

and:

(4 + y)/2 = 7

Solve each equation separately.

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Finding the Missing x-Coordinate

Start with:

(2 + x)/2 = 5

Multiply both sides by 2:

2 + x = 10

Subtract 2:

x = 8

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Finding the Missing y-Coordinate

Now:

(4 + y)/2 = 7

Multiply both sides by 2:

4 + y = 14

Subtract 4:

y = 10

Therefore:

B(8, 10)

Check:

Midpoint of:

(2, 4)

and:

(8, 10)

is:

(5, 7)

Correct.

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A Shortcut for Missing Endpoints

If:

M

is the midpoint between:

A

and:

B

then for the x-coordinate:

x₂ = 2xₘ − x₁

and for the y-coordinate:

y₂ = 2yₘ − y₁

In words:

missing coordinate = 2(midpoint coordinate) − known endpoint coordinate

This shortcut comes directly from rearranging the midpoint formula.

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Worked Example 4: Missing Endpoint

Endpoint:

A(3, 5)

Midpoint:

M(7, 9)

Find B.

For x:

x₂ = 2(7) − 3

x₂ = 14 − 3

x₂ = 11

For y:

y₂ = 2(9) − 5

y₂ = 18 − 5

y₂ = 13

Therefore:

B(11, 13)

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Worked Example 5: Missing Endpoint with Negatives

Endpoint:

A(−4, 6)

Midpoint:

M(2, 1)

Find B.

x-coordinate:

x₂ = 2(2) − (−4)

x₂ = 4 + 4

x₂ = 8

y-coordinate:

y₂ = 2(1) − 6

y₂ = 2 − 6

y₂ = −4

Therefore:

B(8, −4)

---

Another Way to Think About Missing Endpoints

Suppose:

A(2, 3)

and:

M(6, 7)

From A to M, the change is:

Horizontal:

6 − 2 = 4

Vertical:

7 − 3 = 4

Because M is halfway, repeat the same change from M:

Horizontal:

6 + 4 = 10

Vertical:

7 + 4 = 11

Therefore:

B(10, 11)

This method is often very intuitive when working from a graph.

---

Midpoints and Symmetry

The midpoint represents the center of a line segment.

If one endpoint is moved to the other side of the midpoint by exactly the same amount, we reach the second endpoint.

For example:

A(1, 4)

M(5, 7)

From A to M:

+4 horizontally

+3 vertically

Therefore, from M to B:

+4 horizontally

+3 vertically

So:

B(9, 10)

This makes the midpoint a type of center of symmetry for the two endpoints.

---

Midpoints in Geometry

Midpoints are frequently used in geometric figures.

For example, in a triangle, joining the midpoints of two sides creates a:

midsegment

A triangle's midsegment is:

• parallel to the third side
• half the length of the third side

Midpoints are also useful when studying:

• quadrilaterals
• diagonals
• perpendicular bisectors
• transformations
• coordinate proofs

---

Midpoints of Diagonals

An important property of a parallelogram is that its diagonals bisect each other.

This means the diagonals have the same midpoint.

This fact can be used in coordinate geometry to:

• prove that a quadrilateral is a parallelogram
• find missing vertices
• check geometric relationships

---

Worked Example 6: Testing Diagonals

Suppose one diagonal has endpoints:

A(1, 2)

and:

C(7, 8)

Its midpoint is:

((1 + 7)/2, (2 + 8)/2)

= (4, 5)

Another diagonal has endpoints:

B(2, 8)

and:

D(6, 2)

Its midpoint is:

((2 + 6)/2, (8 + 2)/2)

= (4, 5)

Both diagonals have the same midpoint.

This is consistent with the diagonals bisecting each other.

---

Midpoint vs Distance

The distance formula and midpoint formula answer different questions.

Distance asks:

How far apart are the points?

Midpoint asks:

What point lies exactly halfway between them?

For the same two endpoints, we can calculate both.

Both concepts are based on the positions of two coordinate points.

---

Real-World Application: Meeting Halfway

Suppose two friends live at locations:

A(2, 4)

and:

B(10, 8)

If the coordinate system represents a simplified map, a location halfway between them is:

Average x:

(2 + 10)/2 = 6

Average y:

(4 + 8)/2 = 6

Therefore:

M(6, 6)

This point is geographically halfway in the coordinate model.

---

Real-World Application: Emergency Services

Suppose two communities are located at:

A(4, 2)

and:

B(12, 10)

A planner wants to identify the coordinate point exactly halfway between them.

Midpoint:

M((4 + 12)/2, (2 + 10)/2)

M(8, 6)

The midpoint might be one piece of information used when considering the location of a shared facility.

In reality, planners would also need to consider roads, population, terrain, travel times, land availability, and many other factors.

---

Real-World Application: Finding the Centre

Suppose a rectangular sports field has opposite corners:

A(0, 0)

and:

C(100, 60)

The centre of the rectangle is the midpoint of the diagonal.

Average x:

(0 + 100)/2 = 50

Average y:

(0 + 60)/2 = 30

Therefore:

centre = (50, 30)

If the coordinates are measured in metres, this gives the physical centre of the field.

---

Real-World Application: Computer Graphics

Computer graphics use coordinates to position objects on a screen.

Suppose a line is drawn from:

(100, 200)

to:

(500, 400)

The midpoint is:

((100 + 500)/2, (200 + 400)/2)

= (300, 300)

A program could use this point to place:

• a label
• a control point
• a marker
• an animation
• another graphical object

exactly halfway along the line.

---

Real-World Application: Design and Engineering

Engineers and designers often need to identify the centre of components.

If two points describe the ends of a straight component, the midpoint gives its geometric centre.

This can help with:

• positioning supports
• locating sensors
• placing labels
• aligning components
• creating symmetric designs

---

Midpoint and Balance

The word midpoint is sometimes associated with balance because it identifies the geometric centre of a uniform line segment.

For a uniform straight object with evenly distributed mass, the midpoint may also correspond to its centre of mass.

However, this is not always true for real objects.

If mass is distributed unevenly, the centre of mass may not be at the geometric midpoint.

This is an important distinction between geometry and physics.

---

Real-World Application: Mapping

Suppose two locations on a coordinate map are:

P(−6, 4)

and:

Q(8, 12)

Find their midpoint.

Average x:

(−6 + 8)/2 = 1

Average y:

(4 + 12)/2 = 8

Therefore:

M(1, 8)

If each coordinate unit represents 1 km, the point (1, 8) represents the coordinate location halfway along the straight segment joining P and Q.

---

Worked Example 7: Decimal Midpoint

Find the midpoint between:

A(1.2, 3.4)

and:

B(5.8, 7.6)

Average x:

(1.2 + 5.8)/2

= 7/2

= 3.5

Average y:

(3.4 + 7.6)/2

= 11/2

= 5.5

Therefore:

M(3.5, 5.5)

---

Worked Example 8: Geometry Problem

A rectangle has opposite vertices:

A(2, 3)

and:

C(12, 9)

Find the centre of the rectangle.

The centre is the midpoint of diagonal AC.

Average x:

(2 + 12)/2 = 7

Average y:

(3 + 9)/2 = 6

Therefore:

centre = (7, 6)

---

Worked Example 9: Finding an Endpoint Algebraically

The midpoint of AB is:

M(3, −2)

One endpoint is:

A(−5, 4)

Find B.

Let:

B(x, y)

For x:

(−5 + x)/2 = 3

Multiply by 2:

−5 + x = 6

Therefore:

x = 11

For y:

(4 + y)/2 = −2

Multiply by 2:

4 + y = −4

Therefore:

y = −8

So:

B(11, −8)

---

Worked Example 10: Checking the Answer

Check that:

A(−5, 4)

and:

B(11, −8)

have midpoint:

M(3, −2)

Average x:

(−5 + 11)/2 = 6/2 = 3

Average y:

(4 + −8)/2 = −4/2 = −2

Therefore:

M(3, −2)

Correct.

---

Midpoint Problems with Variables

Coordinates can contain variables.

Suppose:

A(2, 4)

and:

B(x, 10)

have midpoint:

M(6, 7)

Use the x-coordinate:

(2 + x)/2 = 6

Multiply by 2:

2 + x = 12

Therefore:

x = 10

The y-coordinate can also be checked:

(4 + 10)/2 = 7

Correct.

---

Using Midpoints to Find Missing Vertices

Suppose a parallelogram has three known vertices and one unknown vertex.

Because the diagonals of a parallelogram bisect each other, we can:

1. find the midpoint of the known diagonal
2. use that midpoint with the known endpoint of the other diagonal
3. calculate the missing endpoint

This combines midpoint calculations with geometric properties.

---

Midpoint and Scale

Coordinates may represent scaled real-world distances.

Suppose:

A(2, 6)

and:

B(14, 18)

The midpoint is:

M(8, 12)

If one coordinate unit represents:

500 m

the coordinate midpoint remains:

(8, 12)

The scale changes the real-world interpretation of the coordinate system, but it does not change how the midpoint is calculated.

---

A Midpoint Is a Point, Not a Distance

This is an important distinction.

Suppose:

A(2, 4)

and:

B(8, 10)

The midpoint is:

(5, 7)

Do not write:

midpoint = 6 units

A midpoint is a location, so it should normally be written as an ordered pair.

Distance is a length.

---

Midpoint vs Average Distance

The midpoint formula averages the coordinates, not the distances from the origin.

For:

A(x₁, y₁)

and:

B(x₂, y₂)

calculate:

average x

and:

average y

separately.

Do not calculate the distance of each point from the origin and then average those distances.

That will not generally give the midpoint.

---

Visualizing the Midpoint

A useful way to understand midpoint problems is to imagine moving from one endpoint toward the other.

Suppose:

A(2, 3)

and:

B(10, 9)

Total horizontal change:

8

Half:

4

Total vertical change:

6

Half:

3

Start at A:

(2 + 4, 3 + 3)

which gives:

(6, 6)

This is the same result produced by the midpoint formula.

---

Fractional Movement Along a Segment

The midpoint represents:

1/2

of the way from one endpoint to another.

If the change from A to B is:

Δx = 12

and:

Δy = 8

then the change from A to the midpoint is:

Δx = 6

and:

Δy = 4

This idea can later be extended to divide a line segment into thirds, quarters, or other ratios.

---

A Reliable Problem-Solving Strategy

For midpoint problems:

1. Identify what is known.

Do you know:

• both endpoints?
• one endpoint and the midpoint?
• a geometric relationship?

2. Write the midpoint relationship.

3. Match x-coordinates with x-coordinates.

4. Match y-coordinates with y-coordinates.

5. Solve each coordinate separately.

6. Write the final answer as an ordered pair.

7. Check that the point is halfway between the endpoints.

---

Common Mistakes

Mistake 1: Adding the coordinates but forgetting to divide by 2

For:

A(2, 4)

and:

B(8, 10)

the midpoint is not:

(10, 14)

You must average the coordinates:

(5, 7)

---

Mistake 2: Mixing x- and y-coordinates

Do not calculate:

(x₁ + y₂)/2

Average:

x with x

and:

y with y

---

Mistake 3: Subtracting coordinates

The standard midpoint formula uses the average, so add the corresponding coordinates and divide by 2.

---

Mistake 4: Forgetting negative signs

For:

−6 + 2

the result is:

−4

Carefully use brackets when necessary.

---

Mistake 5: Assuming midpoint coordinates must be whole numbers

A midpoint such as:

(3.5, 6.5)

is completely valid.

---

Mistake 6: Giving a distance instead of a coordinate

The midpoint is a:

point

so write it as:

(x, y)

---

Mistake 7: Using the midpoint formula incorrectly for a missing endpoint

If one endpoint is missing, create an equation or use:

missing coordinate = 2(midpoint) − known coordinate

---

Mistake 8: Assuming geometric midpoint always means physical balance point

For a uniform object it may, but an uneven distribution of mass can move the centre of mass away from the geometric midpoint.

---

Did You Know?

The midpoint formula is closely related to the idea of an average.

If two numbers are:

a

and:

b

their average is:

(a + b)/2

The midpoint formula simply applies this idea twice:

once to the:

x-coordinates

and once to the:

y-coordinates

In three-dimensional geometry, the same idea can even be extended to three coordinates.

If two points are:

(x₁, y₁, z₁)

and:

(x₂, y₂, z₂)

the midpoint is found by averaging the x-, y-, and z-coordinates separately.

The same simple idea works in higher-dimensional mathematics.

---

Key Terms

• Line segment: Part of a line between two endpoints.
• Endpoint: Point marking the end of a line segment.
• Midpoint: Point exactly halfway between the endpoints of a line segment.
• Coordinate: Number describing a point's position.
• Ordered pair: Coordinates written in the form (x, y).
• Average: Sum of values divided by the number of values.
• Bisect: Divide into two equal parts.
• Segment bisector: Line, ray, segment, or other object that passes through the midpoint and divides a segment into two equal parts.
• Diagonal: Line segment joining non-adjacent vertices of a polygon.
• Parallelogram: Quadrilateral with both pairs of opposite sides parallel.
• Centre: Point representing the middle of a geometric figure.
• Scale: Relationship between coordinates and actual measurements.
• Centre of mass: Point representing the average position of mass in an object.
• Symmetry: Balanced correspondence of positions or shapes around a point, line, or plane.

---

Key Formulas

For endpoints:

A(x₁, y₁)

and:

B(x₂, y₂)

the midpoint is:

M((x₁ + x₂)/2, (y₁ + y₂)/2)

For a missing endpoint:

x₂ = 2xₘ − x₁

y₂ = 2yₘ − y₁

where:

(xₘ, yₘ)

is the midpoint.

---

Key Takeaways

• A midpoint is the point exactly halfway between two endpoints.
• The midpoint divides a line segment into two equal-length parts.
• If M is the midpoint of AB, then AM = MB.
• On a number line, the midpoint is the average of the two endpoints.
• On a coordinate plane, average the x-coordinates and y-coordinates separately.
• The midpoint formula is M((x₁ + x₂)/2, (y₁ + y₂)/2).
• Negative and decimal coordinates do not change the method.
• A midpoint is a coordinate location, not a distance.
• If one endpoint and the midpoint are known, the other endpoint can be calculated.
• A useful missing-endpoint rule is missing coordinate = 2(midpoint coordinate) − known coordinate.
• The midpoint can also be understood as moving halfway through the horizontal and vertical changes between two endpoints.
• Midpoints are useful in geometric problems involving triangles, parallelograms, diagonals, symmetry, and coordinate proofs.
• The diagonals of a parallelogram bisect each other, so they have the same midpoint.
• Midpoint calculations can represent halfway locations on maps and the centres of geometric objects.
• In computer graphics and engineering, midpoint calculations can locate centres, labels, supports, sensors, and control points.
• For a uniform object, the geometric midpoint may correspond to a balance point, although this is not necessarily true when mass is distributed unevenly.
• Understanding the midpoint as an average position makes the formula easier to remember and apply.

3. Segment Relationships

What Is a Line Segment?

A line segment is part of a line with two endpoints.

If the endpoints are A and B, the segment is written:

AB

or symbolically as a segment with endpoints A and B.

Unlike a full line, a segment has a definite length.

On a coordinate plane, we can calculate that length from the coordinates of its endpoints.

---

Segment Length

Suppose a segment has endpoints:

A(2, 3)

and:

B(8, 3)

Because the y-coordinates are equal, AB is horizontal.

Its length is:

AB = |8 − 2|

AB = 6 units

For horizontal and vertical segments, we can often find the length simply by subtracting coordinates.

---

Congruent Line Segments

Two line segments are congruent when they have the same length.

For example:

AB = 5 cm

and:

CD = 5 cm

Therefore:

AB and CD are congruent.

Congruent segments do not need to:

• be in the same location
• point in the same direction
• have the same slope

They only need to have:

equal length

---

Congruence Notation

If segment AB is congruent to segment CD, we can write:

AB ≅ CD

The symbol:

≅

means:

is congruent to

When discussing numerical lengths, we can write:

AB = CD

So there is a useful distinction:

segments are congruent

while:

their lengths are equal

---

Position Does Not Determine Congruence

Consider two horizontal segments.

Segment AB:

A(1, 2) to B(6, 2)

Segment CD:

C(−3, 7) to D(2, 7)

Length AB:

|6 − 1| = 5

Length CD:

|2 − (−3)| = 5

Therefore:

AB ≅ CD

The segments are in different locations, but their lengths are equal.

---

Orientation Does Not Determine Congruence

One segment can be horizontal while another is vertical.

For example:

A(1, 2) and B(6, 2)

give:

AB = 5

Now consider:

C(8, 1) and D(8, 6)

Then:

CD = 5

Therefore:

AB ≅ CD

even though one segment is horizontal and the other is vertical.

---

Measuring Diagonal Segments

If a segment is diagonal, we use the distance formula.

For endpoints:

A(x₁, y₁)

and:

B(x₂, y₂)

the segment length is:

AB = √[(x₂ − x₁)² + (y₂ − y₁)²]

This comes from the Pythagorean Theorem.

---

Why the Distance Formula Works

Suppose:

A(1, 2)

and:

B(5, 5)

Horizontal change:

5 − 1 = 4

Vertical change:

5 − 2 = 3

These form the legs of a right triangle.

Therefore:

AB² = 4² + 3²

AB² = 16 + 9

AB² = 25

So:

AB = 5

The segment has length:

5 units

---

Comparing Two Segment Lengths

To determine whether two segments are congruent:

Step 1: Identify the endpoints of the first segment.

Step 2: Calculate its length.

Step 3: Identify the endpoints of the second segment.

Step 4: Calculate its length.

Step 5: Compare the results.

If:

AB = CD

then:

AB ≅ CD

---

Worked Example 1: Horizontal Segments

Determine whether AB and CD are congruent.

A(2, 3)

B(9, 3)

C(−4, 8)

D(3, 8)

Length AB:

|9 − 2| = 7

Length CD:

|3 − (−4)| = 7

Therefore:

AB = CD = 7

So:

AB ≅ CD

---

Worked Example 2: Horizontal and Vertical

Determine whether these segments are congruent.

A(2, 5)

B(8, 5)

and:

C(10, −1)

D(10, 5)

AB:

|8 − 2| = 6

CD:

|5 − (−1)| = 6

Therefore:

AB ≅ CD

---

Worked Example 3: Diagonal Segments

Consider:

A(1, 1)

B(4, 5)

and:

C(6, 2)

D(9, 6)

Find AB:

AB = √[(4 − 1)² + (5 − 1)²]

AB = √(3² + 4²)

AB = √25

AB = 5

Find CD:

CD = √[(9 − 6)² + (6 − 2)²]

CD = √(3² + 4²)

CD = 5

Therefore:

AB ≅ CD

---

Segments Do Not Need the Same Coordinates

Notice that AB and CD in the previous example had completely different endpoints.

However, both had:

horizontal change = 3

and:

vertical change = 4

Therefore, both had length:

5

This shows that segment congruence depends on:

length

not absolute position.

---

Opposite Direction Does Not Change Length

Suppose segment AB changes:

+4 horizontally

and:

+3 vertically

Another segment CD changes:

−4 horizontally

and:

−3 vertically

Their directions are opposite.

However:

4² = (−4)²

and:

3² = (−3)²

Therefore, both have length:

√(4² + 3²) = 5

Direction does not affect segment length.

---

Worked Example 4: Segments That Are Not Congruent

Consider:

A(0, 0)

B(3, 4)

and:

C(1, 2)

D(7, 2)

First segment:

AB = √(3² + 4²)

AB = 5

Second segment:

CD = |7 − 1|

CD = 6

Therefore:

AB ≠ CD

So the segments are:

not congruent

---

Exact Lengths

Sometimes a segment length is not a whole number.

Consider:

A(1, 2)

and:

B(4, 4)

Then:

AB = √[(4 − 1)² + (4 − 2)²]

AB = √(3² + 2²)

AB = √13

This is an exact length.

If another segment also has length:

√13

the two segments are congruent.

There is no need to convert to a decimal before comparing them.

---

Comparing Squared Distances

Sometimes we can compare segment lengths without calculating the square root.

Suppose:

AB = √(3² + 4²)

and:

CD = √(4² + 3²)

For AB:

AB² = 25

For CD:

CD² = 25

Therefore:

AB = CD

This works because segment lengths are nonnegative.

This technique can make some coordinate proofs faster.

---

Segment Relationships in Geometric Figures

Segment relationships help us study shapes such as:

• triangles
• rectangles
• squares
• parallelograms
• rhombi
• kites

Instead of judging a diagram by appearance, coordinate geometry allows us to calculate whether particular sides or diagonals really have equal lengths.

---

Verifying an Isosceles Triangle

An isosceles triangle has at least two congruent sides.

Suppose:

A(0, 0)

B(6, 0)

C(3, 4)

Calculate AC:

AC = √[(3 − 0)² + (4 − 0)²]

AC = √25

AC = 5

Calculate BC:

BC = √[(3 − 6)² + (4 − 0)²]

BC = √[(-3)² + 4²]

BC = √25

BC = 5

Therefore:

AC ≅ BC

So triangle ABC is isosceles.

---

Verifying an Equilateral Triangle

An equilateral triangle has three congruent sides.

Using coordinates, we can calculate:

AB

BC

and:

CA

If:

AB = BC = CA

then all three sides are congruent.

Therefore, the triangle is equilateral.

This is an example of using coordinate calculations to verify a geometric relationship rather than relying on appearance.

---

Verifying a Rectangle

A rectangle has opposite sides of equal length.

Suppose the vertices are:

A(1, 2)

B(7, 2)

C(7, 6)

D(1, 6)

Lengths:

AB = 6

BC = 4

CD = 6

DA = 4

Therefore:

AB ≅ CD

and:

BC ≅ DA

The opposite sides have equal lengths, as expected.

To fully prove a quadrilateral is a rectangle, however, equal opposite side lengths alone are not enough; additional geometric relationships must also be established.

---

Verifying a Square

A square has four congruent sides.

Suppose:

A(1, 1)

B(5, 1)

C(5, 5)

D(1, 5)

Then:

AB = 4

BC = 4

CD = 4

DA = 4

Therefore:

AB ≅ BC ≅ CD ≅ DA

The four sides have equal length.

A complete coordinate proof that the figure is a square would also verify the necessary angle or perpendicularity relationships.

---

Diagonals

A diagonal connects two non-adjacent vertices of a polygon.

In a rectangle, the two diagonals are congruent.

Using the previous rectangle:

A(1, 2)

B(7, 2)

C(7, 6)

D(1, 6)

Diagonal AC:

AC = √[(7 − 1)² + (6 − 2)²]

AC = √(36 + 16)

AC = √52

Diagonal BD:

BD = √[(1 − 7)² + (6 − 2)²]

BD = √(36 + 16)

BD = √52

Therefore:

AC ≅ BD

---

Segment Bisectors

A segment bisector divides a segment into two congruent segments.

If M is the midpoint of AB, then:

AM ≅ MB

This connects segment relationships with the midpoint concepts from the previous topic.

---

Worked Example 5: Verify a Midpoint

Suppose:

A(2, 3)

M(5, 7)

B(8, 11)

Find AM:

AM = √[(5 − 2)² + (7 − 3)²]

AM = √(9 + 16)

AM = 5

Find MB:

MB = √[(8 − 5)² + (11 − 7)²]

MB = √(9 + 16)

MB = 5

Therefore:

AM ≅ MB

This confirms that M is the midpoint of AB.

---

Using the Midpoint Formula Instead

We can also verify the previous relationship by calculating the midpoint of AB.

A:

(2, 3)

B:

(8, 11)

Midpoint:

((2 + 8)/2, (3 + 11)/2)

= (5, 7)

This matches point M.

So coordinate geometry often gives us more than one method for verifying a relationship.

---

Segment Addition

If point B lies between A and C on the same segment, then:

AB + BC = AC

This is called the Segment Addition Postulate.

For example:

AB = 4 cm

BC = 7 cm

Then:

AC = 4 + 7

AC = 11 cm

---

Finding a Missing Segment Length

If:

AC = 15 cm

and:

AB = 6 cm

then:

AB + BC = AC

So:

6 + BC = 15

Therefore:

BC = 9 cm

This type of relationship can also appear on coordinate diagrams.

---

Worked Example 6: Segment Addition on a Coordinate Plane

Points A, B, and C lie on the same horizontal line:

A(1, 4)

B(5, 4)

C(12, 4)

Then:

AB = 4

BC = 7

and:

AC = 11

Check:

AB + BC = AC

4 + 7 = 11

Correct.

---

Finding an Unknown Coordinate

Suppose:

A(2, 5)

and:

B(x, 5)

The length AB is:

7 units

Because the segment is horizontal:

|x − 2| = 7

There are two possible locations for B:

x = 9

or:

x = −5

This illustrates an important idea:

Knowing a distance does not always determine a unique position.

---

Worked Example 7: Missing Coordinate

Point A is:

A(3, 2)

Point B is:

B(x, 2)

and:

AB = 5

Because the segment is horizontal:

|x − 3| = 5

Therefore:

x = 8

or:

x = −2

So B could be:

(8, 2)

or:

(−2, 2)

unless additional information tells us which side of A contains B.

---

Equal Distances from a Point

Suppose point P is equally distant from points A and B.

Then:

PA = PB

This means:

PA ≅ PB

A point with this property is called equidistant from A and B.

Points on the perpendicular bisector of segment AB are equidistant from A and B.

This relationship becomes important in later geometry topics.

---

Worked Example 8: Equidistant Point

Suppose:

A(0, 0)

B(6, 0)

and:

P(3, 4)

Calculate PA:

PA = √[(3 − 0)² + (4 − 0)²]

PA = 5

Calculate PB:

PB = √[(3 − 6)² + (4 − 0)²]

PB = 5

Therefore:

PA ≅ PB

Point P is equidistant from A and B.

---

Coordinate Proof

A coordinate proof uses algebra and coordinate calculations to establish geometric relationships.

Instead of saying:

"The sides look equal,"

we calculate their lengths.

Instead of saying:

"The point looks like the midpoint,"

we verify its coordinates or distances.

Coordinate proofs can use:

• distance
• midpoint
• slope
• equations of lines

Segment relationships often rely particularly on distance and midpoint.

---

Why Diagrams Can Be Misleading

A geometry diagram may not be drawn perfectly to scale.

Two segments may look equal but have different lengths.

Two segments may look different but actually be congruent.

Therefore:

appearance is not proof

Coordinate geometry allows us to replace visual guessing with calculation.

---

Worked Example 9: Coordinate Verification

A quadrilateral has vertices:

A(0, 0)

B(4, 3)

C(8, 0)

D(4, −3)

Calculate AB:

AB = √(4² + 3²) = 5

Calculate BC:

BC = √(4² + (−3)²) = 5

Calculate CD:

CD = √[(4 − 8)² + (−3 − 0)²] = 5

Calculate DA:

DA = √[(0 − 4)² + (0 − (−3))²] = 5

Therefore:

AB ≅ BC ≅ CD ≅ DA

All four sides are congruent.

This equal-side relationship is consistent with the quadrilateral being a rhombus, though a full classification may require checking other properties depending on what has already been established.

---

Real-World Application: Construction

Suppose a builder is checking two structural supports represented on a coordinate plan.

Support AB:

A(2, 2) to B(8, 10)

Support CD:

C(12, 3) to D(18, 11)

For AB:

Horizontal change = 6

Vertical change = 8

AB = √(6² + 8²) = 10

For CD:

Horizontal change = 6

Vertical change = 8

CD = 10

Therefore, the supports have equal lengths.

---

Real-World Application: Maps

A coordinate map can represent locations such as:

• roads
• buildings
• landmarks
• paths
• utility lines

Suppose two paths connect different locations.

Distance calculations can determine whether the paths have equal straight-line lengths.

The result can then be converted using the map's scale.

---

Real-World Application: Computer Graphics

Computer graphics use coordinates to position objects.

Two line segments might represent:

• edges of a shape
• parts of a character
• structural elements in a model
• sides of a digital object

Distance calculations allow software to determine whether different segments have identical lengths.

This can help maintain:

• symmetry
• proportions
• alignment
• accurate dimensions

---

Real-World Application: Surveying

Surveyors often work with coordinates representing measured positions.

If two pairs of survey points are known, their distances can be calculated.

This allows surveyors and engineers to compare:

• boundaries
• structural dimensions
• distances between reference points
• planned and measured dimensions

Coordinate geometry therefore provides a mathematical connection between location and length.

---

Real-World Application: Sports

Suppose a coach models player positions using coordinates.

Player movement 1:

A(1, 2) to B(7, 10)

Player movement 2:

C(4, 3) to D(10, 11)

Both have:

Horizontal change = 6

Vertical change = 8

Therefore:

distance = 10 units

for both movements.

The movements have equal straight-line lengths even though they begin at different locations.

---

Worked Example 10: Multi-Step Problem

A triangle has vertices:

A(0, 0)

B(8, 0)

C(4, 3)

Determine whether AC and BC are congruent.

Calculate AC:

AC = √[(4 − 0)² + (3 − 0)²]

AC = √(16 + 9)

AC = 5

Calculate BC:

BC = √[(4 − 8)² + (3 − 0)²]

BC = √(16 + 9)

BC = 5

Therefore:

AC ≅ BC

The triangle has two congruent sides.

Therefore, ABC is an:

isosceles triangle

---

Combining Distance and Midpoint

Coordinate geometry problems often require more than one method.

For example, suppose we want to determine whether point M is the midpoint of AB.

We could:

Method 1: Use the midpoint formula

or:

Method 2: Show that M lies on AB and verify AM = MB

This illustrates how coordinate methods can work together to establish geometric relationships.

---

Comparing Segments Efficiently

Suppose:

AB² = 65

and:

CD² = 65

We immediately know:

AB = CD

because both segment lengths are nonnegative.

Therefore:

AB ≅ CD

There is no need to calculate:

√65

as a decimal.

This can make coordinate proofs much quicker.

---

A Reliable Segment Comparison Strategy

When asked whether two segments are congruent:

1. Identify the endpoints.

2. Decide whether each segment is horizontal, vertical, or diagonal.

3. Calculate each length.

4. Keep radical answers exact when possible.

5. Compare the lengths.

6. State the geometric conclusion clearly.

For example:

AB = 5 and CD = 5, therefore AB ≅ CD.

---

A Reliable Coordinate-Proof Strategy

When verifying a geometric relationship:

1. Identify the property that must be shown.

For example:

"An isosceles triangle has two congruent sides."

2. Decide which coordinate method tests that property.

For congruent segments:

use distance

For a bisected segment:

use midpoint and/or distance

3. Perform the calculations.

4. Compare the results.

5. State what the calculations prove.

A calculation without a conclusion is incomplete.

---

Common Mistakes

Mistake 1: Deciding segments are congruent because they look equal

A diagram may not be drawn to scale.

Calculate the lengths.

---

Mistake 2: Thinking congruent segments must have the same slope

They do not.

Congruence requires:

equal length

---

Mistake 3: Thinking congruent segments must be parallel

They do not.

A horizontal and vertical segment can be congruent.

---

Mistake 4: Thinking congruent segments must have the same coordinates

Their positions can be completely different.

Only their lengths need to match.

---

Mistake 5: Adding coordinate differences

For a diagonal segment, do not calculate:

Δx + Δy

Use the distance formula.

---

Mistake 6: Rounding too early

If:

AB = √13

and:

CD = √13

you already know:

AB ≅ CD

There is no reason to introduce rounding error.

---

Mistake 7: Confusing congruent with parallel

Congruent describes equal size or length.

Parallel describes lines with the same direction that never intersect in the plane.

These are different relationships.

---

Mistake 8: Confusing congruent with perpendicular

Perpendicular lines meet at a 90° angle.

This does not tell us whether their lengths are equal.

---

Mistake 9: Assuming equal sides prove every type of shape

Equal side lengths may establish one important property, but a complete classification may require additional information about angles, parallel sides, slopes, or diagonals.

---

Did You Know?

Coordinate geometry provides a powerful way to turn geometric statements into algebraic calculations.

For example:

"These sides are congruent."

can be tested using the:

distance formula

"This point bisects the segment."

can be tested using:

midpoints and distances

"These sides are parallel."

can later be tested using:

slope

"These sides are perpendicular."

can also be investigated using:

slope

This approach is called coordinate geometry or analytic geometry because algebra is used to analyze geometric relationships.

---

Key Terms

• Line segment: Part of a line between two endpoints.
• Endpoint: Point at the end of a line segment.
• Length: Distance between the endpoints of a segment.
• Congruent segments: Line segments with equal lengths.
• Congruence: Relationship in which geometric objects have the same size and shape; for segments, this means equal length.
• Distance formula: Formula used to calculate the length between two coordinate points.
• Midpoint: Point exactly halfway between two endpoints.
• Bisect: Divide into two equal parts.
• Segment bisector: Object that divides a segment into two congruent segments.
• Segment Addition Postulate: If B lies between A and C, then AB + BC = AC.
• Diagonal: Segment connecting non-adjacent vertices of a polygon.
• Equidistant: Being the same distance from two or more points.
• Coordinate proof: Use of coordinate calculations to verify a geometric relationship.
• Isosceles triangle: Triangle with at least two congruent sides.
• Equilateral triangle: Triangle with three congruent sides.
• Parallel: Lines in the same plane that do not intersect.
• Perpendicular: Lines that intersect at a right angle.

---

Key Relationships

For a horizontal segment:

length = |x₂ − x₁|

For a vertical segment:

length = |y₂ − y₁|

For any segment:

length = √[(x₂ − x₁)² + (y₂ − y₁)²]

For congruent segments:

AB = CD

therefore:

AB ≅ CD

If M is the midpoint of AB:

AM = MB

therefore:

AM ≅ MB

For three collinear points with B between A and C:

AB + BC = AC

---

Key Takeaways

• A line segment has two endpoints and a measurable length.
• Two segments are congruent when they have equal lengths.
• Congruent segments do not need to have the same position, direction, or slope.
• Horizontal segment lengths can be found by comparing x-coordinates.
• Vertical segment lengths can be found by comparing y-coordinates.
• Diagonal segment lengths can be calculated using the distance formula.
• Exact radical lengths can often be compared without converting them to decimals.
• Comparing squared distances can make some coordinate proofs more efficient.
• The midpoint of a segment divides it into two congruent segments.
• The Segment Addition Postulate states that if B lies between A and C, then AB + BC = AC.
• Coordinate calculations can verify relationships instead of relying on how a diagram appears.
• Equal side lengths can help verify properties of triangles and quadrilaterals.
• Distance calculations can show that a triangle is isosceles or equilateral.
• Equal side lengths alone may not be enough to completely classify a quadrilateral; other properties may also need to be checked.
• Coordinate methods are useful for solving missing-length and missing-coordinate problems.
• Segment relationships have applications in mapping, construction, engineering, surveying, computer graphics, and sports analysis.
• A strong coordinate proof should show the calculations and clearly state the geometric conclusion.

4. Perimeter and Area on the Coordinate Plane

Geometry on the Coordinate Plane

A coordinate plane allows us to describe geometric figures using the coordinates of their vertices.

For example, a rectangle might have vertices:

A(1, 2)

B(7, 2)

C(7, 6)

D(1, 6)

From these coordinates, we can determine:

• side lengths
• perimeter
• area
• shape properties
• distances between vertices
• relationships between sides

This connects algebra and geometry.

---

Reviewing Side Lengths

Before calculating perimeter or area, we often need to determine the lengths of the sides.

For horizontal sides:

length = |x₂ − x₁|

For vertical sides:

length = |y₂ − y₁|

For diagonal sides, use the distance formula:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Choosing the simplest appropriate method makes calculations faster.

---

Horizontal Side Lengths

Suppose:

A(2, 3)

and:

B(9, 3)

Because the y-coordinates are the same, AB is horizontal.

Therefore:

AB = |9 − 2|

AB = 7 units

---

Vertical Side Lengths

Suppose:

B(9, 3)

and:

C(9, 8)

Because the x-coordinates are the same, BC is vertical.

Therefore:

BC = |8 − 3|

BC = 5 units

---

Diagonal Side Lengths

Suppose:

A(1, 2)

and:

B(5, 5)

Horizontal change:

5 − 1 = 4

Vertical change:

5 − 2 = 3

Therefore:

AB = √(4² + 3²)

AB = √25

AB = 5 units

---

What Is Perimeter?

The perimeter of a polygon is the total distance around its outside boundary.

To calculate perimeter:

add the lengths of all sides

For a polygon with side lengths:

a, b, c, d

the perimeter is:

P = a + b + c + d

The exact formula depends on the shape, but the basic idea is always the same:

Perimeter = total boundary length

---

Perimeter Units

Because perimeter measures length, it uses linear units.

Examples include:

• mm
• cm
• m
• km

If no measurement unit is given, write:

units

For example:

P = 24 units

---

Perimeter of a Rectangle

Consider the rectangle:

A(1, 2)

B(7, 2)

C(7, 6)

D(1, 6)

Find AB:

AB = 7 − 1 = 6

Find BC:

BC = 6 − 2 = 4

Opposite sides of a rectangle are equal, so:

CD = 6

DA = 4

Perimeter:

P = 6 + 4 + 6 + 4

P = 20 units

We could also use:

P = 2l + 2w

P = 2(6) + 2(4)

P = 20 units

---

What Is Area?

Area measures the amount of two-dimensional space inside a figure.

Perimeter measures the:

boundary

Area measures the:

region inside the boundary

This distinction is very important.

---

Area Units

Area uses square units.

Examples:

• mm²
• cm²
• m²
• km²

If no measurement unit is specified, write:

square units

For example:

A = 24 square units

or:

A = 24 units²

---

Area of a Rectangle

The formula for the area of a rectangle is:

A = length × width

For our previous rectangle:

Length:

6 units

Width:

4 units

Therefore:

A = 6 × 4

A = 24 units²

Notice:

Perimeter = 20 units

but:

Area = 24 units²

They measure different things.

---

Perimeter vs Area

A common mistake is confusing perimeter and area.

Perimeter

asks:

"How far is it around the figure?"

Area

asks:

"How much space is inside the figure?"

Think of:

Perimeter → fence

Area → floor

If you were fencing a field, you would need its perimeter.

If you were covering the field with grass, you would need its area.

---

A Reliable Coordinate Perimeter Method

When finding the perimeter of a coordinate figure:

Step 1: Identify the vertices in order.

Step 2: Determine which points are connected.

Step 3: Calculate every side length.

Step 4: Use simple subtraction for horizontal or vertical sides.

Step 5: Use the distance formula for diagonal sides.

Step 6: Add all side lengths.

Step 7: Include the correct units.

---

Worked Example 1: Rectangle

A rectangle has vertices:

A(2, 1)

B(10, 1)

C(10, 6)

D(2, 6)

Length:

10 − 2 = 8

Width:

6 − 1 = 5

Perimeter:

P = 2(8) + 2(5)

P = 16 + 10

P = 26 units

Area:

A = 8 × 5

A = 40 units²

---

Worked Example 2: Square

A square has vertices:

A(−2, 1)

B(4, 1)

C(4, 7)

D(−2, 7)

Side length:

4 − (−2) = 6

Therefore:

P = 4(6)

P = 24 units

Area:

A = 6²

A = 36 units²

---

Area of a Triangle

The area of a triangle is:

A = ½bh

where:

• b = base
• h = perpendicular height

The height must meet the base at a:

90° angle

It is not necessarily one of the triangle's sides.

---

Triangle on a Coordinate Plane

Consider:

A(1, 2)

B(7, 2)

C(4, 6)

AB is horizontal.

Base:

7 − 1 = 6

The vertical distance from C to the line AB is:

6 − 2 = 4

Therefore:

A = ½(6)(4)

A = 12 units²

---

Perimeter of the Same Triangle

For:

A(1, 2)

B(7, 2)

C(4, 6)

We already know:

AB = 6

Now find AC:

Horizontal change:

4 − 1 = 3

Vertical change:

6 − 2 = 4

Therefore:

AC = √(3² + 4²)

AC = 5

Similarly:

BC = 5

So:

P = 6 + 5 + 5

P = 16 units

The triangle has:

Perimeter = 16 units

Area = 12 units²

---

Area Does Not Require Every Side Length

Notice that we needed all three side lengths to calculate the triangle's perimeter.

But for its area, we only needed:

• base
• perpendicular height

This is an important problem-solving idea.

Do not calculate information that you do not need.

---

Right Triangles

Coordinate planes make the area of some right triangles especially easy to calculate.

Suppose:

A(2, 2)

B(8, 2)

C(2, 7)

AB is horizontal:

AB = 6

AC is vertical:

AC = 5

These sides are perpendicular.

Therefore:

A = ½(6)(5)

A = 15 units²

---

Perimeter of a Right Triangle

Using the same triangle:

AB = 6

AC = 5

Find BC:

BC = √(6² + 5²)

BC = √61

Therefore:

P = 6 + 5 + √61

Exact perimeter:

P = 11 + √61 units

Since:

√61 ≈ 7.81

Approximate perimeter:

P ≈ 18.81 units

---

Area of a Parallelogram

The area of a parallelogram is:

A = bh

where:

• b = base
• h = perpendicular height

The slanted side is not the height unless it is perpendicular to the base.

---

Worked Example 3: Parallelogram

Consider:

A(1, 2)

B(7, 2)

C(9, 6)

D(3, 6)

Base AB:

7 − 1 = 6

Vertical height:

6 − 2 = 4

Therefore:

A = 6 × 4

A = 24 units²

Notice that the slanted side AD is not used as the height.

---

Perimeter of the Parallelogram

AB:

6

AD has:

Horizontal change:

3 − 1 = 2

Vertical change:

6 − 2 = 4

Therefore:

AD = √(2² + 4²)

AD = √20

AD = 2√5

Opposite sides are equal.

Therefore:

P = 2(6) + 2(2√5)

P = 12 + 4√5 units

---

Area of a Trapezoid

The area of a trapezoid is:

A = ½(b₁ + b₂)h

where:

• b₁ and b₂ are the parallel sides
• h is the perpendicular distance between them

---

Worked Example 4: Trapezoid

Suppose the vertices are:

A(1, 2)

B(9, 2)

C(7, 6)

D(3, 6)

The parallel bases are:

AB = 8

and:

CD = 4

Height:

6 − 2 = 4

Therefore:

A = ½(8 + 4)(4)

A = ½(12)(4)

A = 24 units²

---

Finding the Trapezoid's Perimeter

Using the same trapezoid:

AB = 8

CD = 4

For BC:

Horizontal change:

9 − 7 = 2

Vertical change:

6 − 2 = 4

So:

BC = √(2² + 4²)

BC = √20

BC = 2√5

Similarly:

DA = 2√5

Therefore:

P = 8 + 4 + 2√5 + 2√5

P = 12 + 4√5 units

---

Irregular Coordinate Figures

Not every coordinate polygon is a simple rectangle, triangle, or trapezoid.

For an irregular figure, we can often divide it into simpler shapes.

For example, an irregular polygon might be divided into:

• rectangles
• triangles
• trapezoids

Calculate the area of each part and then add the results.

---

Composite Figures

A composite figure is made from two or more simpler shapes.

Suppose an L-shaped figure can be divided into two rectangles.

Rectangle 1:

8 × 3 = 24

Rectangle 2:

4 × 2 = 8

Total area:

24 + 8 = 32 units²

Another method is to calculate the area of a large rectangle and subtract a missing rectangle.

Both methods should give the same result.

---

Subtraction Method

Suppose a shape fits inside a:

10 × 8 rectangle

but has a:

4 × 3 rectangle

removed.

Large rectangle:

10 × 8 = 80

Missing rectangle:

4 × 3 = 12

Area of shape:

80 − 12 = 68 units²

---

Be Careful with Composite Perimeters

Area and perimeter behave differently when shapes are combined.

For area:

add the areas of the pieces

But for perimeter, only add the lengths on the outside boundary.

Do not include interior lines used to divide the figure.

This is a common source of mistakes.

---

Using Coordinates to Identify Shapes

Coordinates can tell us more than perimeter and area.

We can calculate side lengths to investigate whether a figure might be:

• square
• rectangle
• rhombus
• parallelogram
• isosceles triangle
• equilateral triangle

For example, if all four sides have equal length, the figure has an important property of a rhombus.

However, side lengths alone may not always be enough to completely classify the figure.

---

Worked Example 5: Analyze a Quadrilateral

Consider:

A(0, 0)

B(6, 0)

C(6, 4)

D(0, 4)

Side lengths:

AB = 6

BC = 4

CD = 6

DA = 4

The sides are horizontal and vertical, so adjacent sides meet at right angles.

Therefore, the figure is a:

rectangle

Perimeter:

P = 6 + 4 + 6 + 4

P = 20 units

Area:

A = 6 × 4

A = 24 units²

---

Worked Example 6: Square or Rectangle?

Consider:

A(−2, −2)

B(3, −2)

C(3, 3)

D(−2, 3)

Each side has length:

5 units

The sides are horizontal and vertical, so adjacent sides are perpendicular.

Therefore, the figure is a:

square

Perimeter:

P = 4(5) = 20 units

Area:

A = 5² = 25 units²

---

Rotated Figures

A shape does not have to have horizontal and vertical sides.

Consider a square that has been rotated on the coordinate plane.

Its side lengths may need to be found using the distance formula.

We cannot assume that:

horizontal difference = side length

when the side is diagonal.

---

Worked Example 7: Rotated Square

Consider the vertices:

A(0, 3)

B(3, 6)

C(6, 3)

D(3, 0)

Find AB:

AB = √[(3 − 0)² + (6 − 3)²]

AB = √(9 + 9)

AB = √18

AB = 3√2

All four sides have this length.

Therefore:

P = 4(3√2)

P = 12√2 units

---

Area of the Rotated Square

The diagonals are easy to measure.

AC extends from:

(0, 3) to (6, 3)

So:

AC = 6

BD extends from:

(3, 6) to (3, 0)

So:

BD = 6

For this square, the area can be calculated from the diagonals:

A = ½d₁d₂

Therefore:

A = ½(6)(6)

A = 18 units²

This agrees with using:

side² = (3√2)² = 18

---

Coordinate Area by Counting Squares

When a figure lies on a coordinate grid, we can sometimes estimate or calculate area by counting unit squares.

A single:

1 × 1

square has area:

1 square unit

This method is particularly useful for:

• visualizing area
• checking calculations
• simple rectangles
• simple composite figures

---

Area by Enclosing Rectangle

Another useful method for irregular figures is to draw the smallest convenient rectangle around the polygon.

Then:

Area of polygon = Area of enclosing rectangle − Area outside polygon

The outside regions can often be divided into triangles or rectangles.

This method is particularly useful for slanted coordinate figures.

---

Worked Example 8: Enclosing Rectangle

Suppose a triangle lies inside a:

8 × 6

rectangle.

Area of rectangle:

8 × 6 = 48

Suppose the three regions outside the triangle have areas:

6

10

and:

8

Then:

Triangle area = 48 − 6 − 10 − 8

Triangle area = 24 units²

This approach can sometimes be easier than finding a perpendicular height directly.

---

Coordinate Geometry and Scale

Coordinate diagrams often represent real objects using a scale.

Suppose:

1 coordinate unit = 10 m

A rectangular property has coordinate dimensions:

8 units × 5 units

Actual dimensions:

80 m × 50 m

Perimeter:

P = 2(80) + 2(50)

P = 260 m

Area:

A = 80 × 50

A = 4000 m²

---

Scale and Area

Be especially careful when converting area.

If:

1 coordinate unit = 5 m

then one square coordinate unit represents:

5 m × 5 m

which equals:

25 m²

Therefore:

1 unit² = 25 m²

not:

5 m²

This is because area involves two dimensions.

---

Worked Example 9: Scale and Area

A coordinate rectangle measures:

6 units × 4 units

and:

1 unit = 3 m

Actual dimensions:

18 m × 12 m

Area:

A = 18 × 12

A = 216 m²

Alternatively:

Coordinate area:

6 × 4 = 24 units²

Since:

1 unit² = 3² = 9 m²

then:

24 × 9 = 216 m²

Both methods give the same answer.

---

Real-World Application: Fencing

Suppose a rectangular garden has coordinate vertices:

(1, 2)

(11, 2)

(11, 8)

(1, 8)

If each coordinate unit represents:

1 metre

then:

Length:

10 m

Width:

6 m

Perimeter:

P = 2(10) + 2(6)

P = 32 m

The garden requires:

32 m of fencing

if the entire boundary is fenced.

---

Real-World Application: Grass

Using the same garden:

Length:

10 m

Width:

6 m

Area:

A = 10 × 6

A = 60 m²

Therefore:

• fencing depends on perimeter
• grass coverage depends on area

This demonstrates why distinguishing the two quantities matters.

---

Real-World Application: Flooring

Suppose a room is represented by a rectangle with dimensions:

8 m × 5 m

Flooring covers the interior.

Therefore, we need:

area

A = 8 × 5

A = 40 m²

If trim is placed around the edges of the room, we need:

perimeter

P = 2(8) + 2(5)

P = 26 m

---

Real-World Application: Land Surveying

Coordinate geometry can be used to represent property boundaries.

Surveyors may work with coordinates that identify corners of a parcel of land.

From these coordinates, mathematical methods can help determine:

• boundary lengths
• perimeter
• area
• distances between points

For irregular parcels, more advanced coordinate-area methods may be used.

---

Real-World Application: Computer Graphics

Digital shapes can be represented by vertices stored as coordinates.

Software can use these coordinates to calculate:

• edge lengths
• perimeter
• area
• centres
• collisions
• dimensions

Coordinate geometry therefore has important applications in:

• computer graphics
• game development
• CAD
• engineering
• digital mapping

---

Real-World Application: Architecture

Architectural drawings often use scaled coordinate-like systems.

A floor plan may represent:

• walls
• rooms
• doors
• windows
• floor areas

Perimeter calculations can help determine quantities such as:

• trim
• fencing
• wall boundaries

Area calculations can help determine quantities such as:

• flooring
• tiles
• carpet
• paint coverage for flat surfaces
• usable floor space

---

Worked Example 10: Practical Coordinate Problem

A rectangular playground has vertices:

A(2, 3)

B(14, 3)

C(14, 11)

D(2, 11)

Each coordinate unit represents:

2 m

Coordinate length:

14 − 2 = 12 units

Actual length:

12 × 2 = 24 m

Coordinate width:

11 − 3 = 8 units

Actual width:

8 × 2 = 16 m

Perimeter:

P = 2(24) + 2(16)

P = 80 m

Area:

A = 24 × 16

A = 384 m²

Therefore:

• 80 m of boundary surrounds the playground.
• The playground covers 384 m².

---

Perimeter of Irregular Polygons

For an irregular polygon, calculate every outside side individually.

Suppose a pentagon has side lengths:

4

5

√13

7

and:

6

Then:

P = 4 + 5 + √13 + 7 + 6

P = 22 + √13 units

If a decimal is required:

√13 ≈ 3.61

Therefore:

P ≈ 25.61 units

---

Keep Exact Values When Possible

If a diagonal side has length:

√20

do not immediately round it.

Simplify:

√20 = √(4 × 5)

√20 = 2√5

Keeping exact values prevents rounding errors.

Only convert to a decimal when required.

---

Using Coordinates to Check a Shape

Suppose a quadrilateral appears to be a square.

A coordinate analysis could include:

Step 1: Calculate all four side lengths.

Step 2: Check whether they are equal.

Step 3: Check whether adjacent sides are perpendicular, if required.

Step 4: Calculate the perimeter.

Step 5: Calculate the area.

This is more reliable than judging the figure by appearance.

---

Choosing the Correct Area Formula

Before calculating area, identify the shape.

Rectangle

A = lw

Square

A = s²

Triangle

A = ½bh

Parallelogram

A = bh

Trapezoid

A = ½(b₁ + b₂)h

For composite or irregular figures:

divide the figure into simpler shapes

or:

subtract unwanted areas from a larger shape

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A Reliable Area Strategy

Step 1: Plot or inspect the vertices.

Step 2: Identify the shape or divide it into simpler shapes.

Step 3: Determine the required dimensions from the coordinates.

Step 4: Choose the appropriate area formula.

Step 5: Calculate carefully.

Step 6: Use square units.

Step 7: Check whether the result is reasonable.

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Checking Perimeter Answers

Ask:

Did I include every outside side?

Did I accidentally include a diagonal that is not part of the boundary?

Did I use the distance formula for diagonal sides?

Did I use linear units?

These questions catch many common mistakes.

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Checking Area Answers

Ask:

Did I identify the correct shape?

Did I use perpendicular height rather than a slanted side?

Did I include all parts of a composite figure?

Did I accidentally count overlapping regions twice?

Did I use square units?

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Common Mistakes

Mistake 1: Confusing perimeter and area

Perimeter measures the boundary.

Area measures the interior.

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Mistake 2: Using square units for perimeter

Perimeter:

m

Area:

m²

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Mistake 3: Using linear units for area

An area of:

24 m

is incorrectly labeled.

It should be:

24 m²

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Mistake 4: Adding coordinates instead of finding side lengths

Coordinates represent positions.

You must determine the distance between the positions.

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Mistake 5: Using simple subtraction for a diagonal side

For diagonal segments, use:

distance formula

unless the length is otherwise known.

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Mistake 6: Using a slanted side as triangle height

The height must be:

perpendicular to the base

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Mistake 7: Using a parallelogram's slanted side as its height

Again, the height must be perpendicular to the base.

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Mistake 8: Including interior lines in a composite perimeter

Only count the:

outside boundary

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Mistake 9: Forgetting to square the scale factor for area

If:

1 unit = 4 m

then:

1 unit² = 16 m²

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Mistake 10: Rounding side lengths too early

Keep exact radical values until the final step whenever possible.

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Did You Know?

Coordinate geometry makes it possible for computers to calculate the area and perimeter of complicated shapes.

Modern applications include:

• geographic information systems (GIS)
• digital maps
• architecture
• land surveying
• computer-aided design
• robotics
• engineering
• computer graphics
• game development

A complicated boundary can be stored as a series of coordinate points.

Mathematical algorithms can then calculate its dimensions.

The same coordinate principles used in school geometry form the foundation for much more advanced applications.

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Key Terms

• Coordinate plane: Two-dimensional system used to locate points.
• Vertex: Point where two sides of a polygon meet.
• Polygon: Closed two-dimensional figure made from straight line segments.
• Side length: Distance between adjacent vertices.
• Perimeter: Total distance around the boundary of a figure.
• Area: Amount of two-dimensional space inside a figure.
• Distance formula: Formula used to calculate distance between coordinate points.
• Base: Chosen side of a figure used in an area calculation.
• Height: Perpendicular distance from a base to the opposite side or vertex.
• Composite figure: Figure made from two or more simpler shapes.
• Scale: Relationship between measurements on a representation and actual measurements.
• Square unit: Unit used for area.
• Diagonal: Segment joining two non-adjacent vertices of a polygon.
• Exact value: Value that has not been rounded.
• Approximation: Rounded value close to the exact value.

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Key Formulas

Horizontal length:

|x₂ − x₁|

Vertical length:

|y₂ − y₁|

Distance formula:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Rectangle perimeter:

P = 2l + 2w

Rectangle area:

A = lw

Square perimeter:

P = 4s

Square area:

A = s²

Triangle area:

A = ½bh

Parallelogram area:

A = bh

Trapezoid area:

A = ½(b₁ + b₂)h

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Key Takeaways

• Coordinates can be used to determine the dimensions of geometric figures.
• Horizontal side lengths are found using differences between x-coordinates.
• Vertical side lengths are found using differences between y-coordinates.
• Diagonal side lengths can be calculated using the distance formula.
• Perimeter is the total distance around a polygon.
• Perimeter uses linear units such as cm, m, or km.
• Area measures the amount of space inside a two-dimensional figure.
• Area uses square units such as cm², m², or km².
• Perimeter and area measure different properties and should not be confused.
• Rectangles and squares can often be analyzed directly from coordinate differences.
• Triangle area requires a base and perpendicular height.
• Parallelogram area also uses perpendicular height rather than the length of a slanted side.
• Trapezoid area depends on the two parallel bases and their perpendicular separation.
• Irregular coordinate figures can often be divided into simpler shapes.
• An enclosing rectangle can sometimes be used to calculate an irregular area by subtraction.
• Composite perimeters include only the outside boundary, not interior dividing lines.
• Exact radical values should usually be kept until the final stage of a calculation.
• Coordinate calculations can also help identify and analyze geometric shapes.
• Scale factors must be interpreted carefully: if lengths are multiplied by a scale factor, areas are multiplied by the square of that scale factor.
• Coordinate perimeter and area calculations have practical applications in construction, architecture, mapping, surveying, landscaping, computer graphics, engineering, and land measurement.

5. Coordinate Geometry Applications