The Midpoint Formula

What is a Midpoint?

The midpoint is the point exactly halfway between two endpoints.

Properties:

• Divides segment into two equal parts
• Same distance from each endpoint

The Midpoint Formula

Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Where:

• (x₁, y₁) = first endpoint
• (x₂, y₂) = second endpoint
• M = midpoint

In words: Average the x-coordinates, average the y-coordinates!

Using the Midpoint Formula

Example 1: Basic Midpoint

Find midpoint of segment from (2, 4) to (8, 10)

Step 1: Identify coordinates

• (x₁, y₁) = (2, 4)
• (x₂, y₂) = (8, 10)

Step 2: Apply formula

M = ((2 + 8)/2, (4 + 10)/2)
M = (10/2, 14/2)
M = `(5, 7)`

Answer: Midpoint is (5, 7)

Check: Distance from (2,4) to (5,7) = Distance from (5,7) to (8,10) ✓

Example 2: Negative Coordinates

Find midpoint of (−3, 5) and (7, −1)

Apply formula:

M = ((−3 + 7)/2, (5 + (−1))/2)
M = (4/2, 4/2)
M = `(2, 2)`

Answer: (2, 2)

Example 3: Origin as Endpoint

Find midpoint of (0, 0) and (6, 8)

Apply formula:

M = ((0 + 6)/2, (0 + 8)/2)
M = `(3, 4)`

Answer: (3, 4)

Shortcut: When one point is origin, midpoint is half of the other point!

Example 4: Fractional Midpoint

Find midpoint of (1, 3) and (4, 6)

Calculate:

M = ((1 + 4)/2, (3 + 6)/2)
M = (5/2, 9/2)
M = (2.5, 4.5)

Answer: (2.5, 4.5) or (5/2, 9/2)

Finding an Endpoint

If you know one endpoint and the midpoint, find the other endpoint!

Strategy: Use algebra

Formula rearranged:

• x₂ = 2M_x − x₁
• y₂ = 2M_y − y₁

Example 1: Find Missing Endpoint

Given:

• Endpoint A: (2, 5)
• Midpoint M: (6, 8)
• Find endpoint B

Use formula:

x₂ = 2(6) − 2 = 12 − 2 = 10
y₂ = 2(8) − 5 = 16 − 5 = 11

Answer: B = (10, 11)

Check: Midpoint of (2,5) and (10,11) = ((2+10)/2, (5+11)/2) = (6, 8) ✓

Example 2: Missing Endpoint with Negatives

Given:

• Endpoint: (−4, 7)
• Midpoint: (2, 3)
• Find other endpoint

Calculate:

x₂ = 2(2) − (−4) = 4 + 4 = 8
y₂ = 2(3) − 7 = 6 − 7 = −1

Answer: (8, −1)

Midpoint on Number Line

For points on a number line (1-dimensional):

Formula: M = (a + b)/2

Example: Number Line

Find midpoint between −5 and 11

Calculate: M = (−5 + 11)/2 = 6/2 = 3

Answer: 3

Trisection Points

Trisection points divide a segment into three equal parts.

Not the same as midpoint!

Example: Find Trisection Points

Segment from (0, 0) to (9, 6)

First trisection point (1/3 of the way):

• x = 0 + (1/3)(9 − 0) = 3
• y = 0 + (1/3)(6 − 0) = 2
• Point: (3, 2)

Second trisection point (2/3 of the way):

• x = 0 + (2/3)(9) = 6
• y = 0 + (2/3)(6) = 4
• Point: (6, 4)

Midpoint would be: (4.5, 3)

Geometric Applications

Example: Rectangle Diagonals

Rectangle vertices: A(1, 2), B(7, 2), C(7, 6), D(1, 6)

Diagonal AC: From (1, 2) to (7, 6) Diagonal BD: From (7, 2) to (1, 6)

Midpoint of AC:

M₁ = ((1 + 7)/2, (2 + 6)/2) = `(4, 4)`

Midpoint of BD:

M₂ = ((7 + 1)/2, (2 + 6)/2) = `(4, 4)`

Observation: Both diagonals have the same midpoint!

Property: Rectangle diagonals bisect each other.

Finding Center of a Shape

Example: Center of Line Segment

Endpoints: (−2, 3) and (6, 11)

Center (midpoint):

C = ((−2 + 6)/2, (3 + 11)/2)
C = `(2, 7)`

Answer: Center at (2, 7)

Real-World Applications

Construction: Find center point for symmetry

• Building layouts, artwork

Navigation: Find halfway point on a route

• Meeting point between two locations

Computer graphics: Bezier curves and midpoints

• Animation, design software

Sports: Place markers at midfield

• Soccer, football field markings

Geography: Find center between two cities

• Population centers, service areas

Midpoint vs. Distance

Different formulas, different purposes!

Midpoint: Finds location halfway between

• M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Distance: Finds length of segment

• d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Example: Both Calculations

Points: A(1, 2) and B(7, 10)

Midpoint:

M = ((1 + 7)/2, (2 + 10)/2) = `(4, 6)`

Distance:

d = √[(7 − 1)² + (10 − 2)²]
d = √[36 + 64] = √100 = 10

Practice