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.

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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

---

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)

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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.

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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.

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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

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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)

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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)

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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)

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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)

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Mistake 2: Mixing x- and y-coordinates

Do not calculate:

(x₁ + y₂)/2

Average:

x with x

and:

y with y

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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.

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Mistake 5: Assuming midpoint coordinates must be whole numbers

A midpoint such as:

(3.5, 6.5)

is completely valid.

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Mistake 6: Giving a distance instead of a coordinate

The midpoint is a:

point

so write it as:

(x, y)

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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

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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.

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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.

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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.

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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.

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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.