- Algebra
- Arithmetic
- Whole Numbers
- Numbers
- Types of Numbers
- Odd and Even Numbers
- Prime & Composite Numbers
- Sieve of Eratosthenes
- Number Properties
- Commutative Property
- Associative Property
- Identity Property
- Distributive Property
- Order of Operations
- Rounding Numbers
- Absolute Value
- Number Sequences
- Factors & Multiples
- Prime Factorization
- Greatest Common Factor
- Least Common Multiple
- Squares & Perfect Squares
- Square Roots
- Squares & Square Roots
- Simplifying Square Roots
- Simplifying Radicals
- Radicals that have Fractions
- Multiplying Radicals

- Integers
- Fractions
- Introducing Fractions
- Converting Fractions
- Comparing Fractions
- Ordering Fractions
- Equivalent Fractions
- Reducing Fractions
- Adding Fractions
- Subtracting Fractions
- Multiplying Fractions
- Reciprocals
- Dividing Fractions
- Adding Mixed Numbers
- Subtracting Mixed Numbers
- Multiplying Mixed Numbers
- Dividing Mixed Numbers
- Complex Fractions
- Fractions to Decimals

- Decimals
- Exponents
- Percent
- Scientific Notation
- Proportions
- Equality
- Properties of equality
- Addition property of equality
- Transitive property of equality
- Subtraction property of equality
- Multiplication property of equality
- Division property of equality
- Symmetric property of equality
- Reflexive property of equality
- Substitution property of equality
- Distributive property of equality

- Commercial Math

- Calculus
- Differential Calculus
- Limits calculus
- Mean value theorem
- L’Hôpital’s rule
- Newton’s method
- Derivative calculus
- Power rule
- Sum rule
- Difference rule
- Product rule
- Quotient rule
- Chain rule
- Derivative rules
- Trigonometric derivatives
- Inverse trig derivatives
- Trigonometric substitution
- Derivative of arctan
- Derivative of secx
- Derivative of csc
- Derivative of cotx
- Exponential derivative
- Derivative of ln
- Implicit differentiation
- Critical numbers
- Derivative test
- Concavity calculus
- Related rates
- Curve sketching
- Asymptote
- Hyperbolic functions
- Absolute maximum
- Absolute minimum

- Integral Calculus
- Fundamental theorem of calculus
- Approximating integrals
- Riemann sum
- Integral properties
- Antiderivative
- Integral calculus
- Improper integrals
- Integration by parts
- Partial fractions
- Area under the curve
- Area between two curves
- Center of mass
- Work calculus
- Integrating exponential functions
- Integration of hyperbolic functions
- Integrals of inverse trig functions
- Disk method
- Washer method
- Shell method

- Sequences, Series & Tests
- Parametric Curves & Polar Coordinates
- Multivariable Calculus
- 3d coordinate system
- Vector calculus
- Vectors equation of a line
- Equation of a plane
- Intersection of line and plane
- Quadric surfaces
- Spherical coordinates
- Cylindrical coordinates
- Vector function
- Derivatives of vectors
- Length of a vector
- Partial derivatives
- Tangent plane
- Directional derivative
- Lagrange multipliers
- Double integrals
- Iterated integral
- Double integrals in polar coordinates
- Triple integral
- Change of variables in multiple integrals
- Vector fields
- Line integral
- Fundamental theorem for line integrals
- Green’s theorem
- Curl vector field
- Surface integral
- Divergence of a vector field
- Differential equations
- Exact equations
- Integrating factor
- First order linear differential equation
- Second order homogeneous differential equation
- Non homogeneous differential equation
- Homogeneous differential equation
- Characteristic equations
- Laplace transform
- Inverse laplace transform
- Dirac delta function

- Differential Calculus
- Matrices
- Pre-Calculus
- Lines & Planes
- Functions
- Domain of a function
- Transformation Of Graph
- Polynomials
- Graphs of rational functions
- Limits of a function
- Complex Numbers
- Exponential Function
- Logarithmic Function
- Sequences
- Conic Sections
- Series
- Mathematical induction
- Probability
- Advanced Trigonometry
- Vectors
- Polar coordinates

- Probability
- Geometry
- Angles
- Triangles
- Types of Triangles
- Special Right Triangles
- 3 4 5 Triangle
- 45 45 90 Triangle
- 30 60 90 Triangle
- Area of Triangle
- Pythagorean Theorem
- Pythagorean Triples
- Congruent Triangles
- Hypotenuse Leg (HL)
- Similar Triangles
- Triangle Inequality
- Triangle Sum Theorem
- Exterior Angle Theorem
- Angles of a Triangle
- Law of Sines or Sine Rule
- Law of Cosines or Cosine Rule

- Polygons
- Circles
- Circle Theorems
- Solid Geometry
- Volume of Cubes
- Volume of Rectangular Prisms
- Volume of Prisms
- Volume of Cylinders
- Volume of Spheres
- Volume of Cones
- Volume of Pyramids
- Volume of Solids
- Surface Area of a Cube
- Surface Area of a Cuboid
- Surface Area of a Prism
- Surface Area of a Cylinder
- Surface Area of a Cone
- Surface Area of a Sphere
- Surface Area of a Pyramid
- Geometric Nets
- Surface Area of Solids

- Coordinate Geometry and Graphs
- Coordinate Geometry
- Coordinate Plane
- Slope of a Line
- Equation of a Line
- Forms of Linear Equations
- Slopes of Parallel and Perpendicular Lines
- Graphing Linear Equations
- Midpoint Formula
- Distance Formula
- Graphing Inequalities
- Linear Programming
- Graphing Quadratic Functions
- Graphing Cubic Functions
- Graphing Exponential Functions
- Graphing Reciprocal Functions

- Geometric Constructions
- Geometric Construction
- Construct a Line Segment
- Construct Perpendicular Bisector
- Construct a Perpendicular Line
- Construct Parallel Lines
- Construct a 60° Angle
- Construct an Angle Bisector
- Construct a 30° Angle
- Construct a 45° Angle
- Construct a Triangle
- Construct a Parallelogram
- Construct a Square
- Construct a Rectangle
- Locus of a Moving Point

- Geometric Transformations

- Sets & Set Theory
- Statistics
- Collecting and Summarizing Data
- Common Ways to Describe Data
- Different Ways to Represent Data
- Frequency Tables
- Cumulative Frequency
- Advance Statistics
- Sample mean
- Population mean
- Sample variance
- Standard deviation
- Random variable
- Probability density function
- Binomial distribution
- Expected value
- Poisson distribution
- Normal distribution
- Bernoulli distribution
- Z-score
- Bayes theorem
- Normal probability plot
- Chi square
- Anova test
- Central limit theorem
- Sampling distribution
- Logistic equation
- Chebyshev’s theorem

- Difference
- Correlation Coefficient
- Tautology
- Relative Frequency
- Frequency Distribution
- Dot Plot
- Сonditional Statement
- Converse Statement
- Law of Syllogism
- Counterexample
- Least Squares
- Law of Detachment
- Scatter Plot
- Linear Graph
- Arithmetic Mean
- Measures of Central Tendency
- Discrete Data
- Weighted Average
- Summary Statistics
- Interquartile Range
- Categorical Data

- Trigonometry
- Vectors
- Multiplication Charts
- Time Table
- 2 times table
- 3 times table
- 4 times table
- 5 times table
- 6 times table
- 7 times table
- 8 times table
- 9 times table
- 10 times table
- 11 times table
- 12 times table
- 13 times table
- 14 times table
- 15 times table
- 16 times table
- 17 times table
- 18 times table
- 19 times table
- 20 times table
- 21 times table
- 22 times table
- 23 times table
- 24 times table

- Time Table

# Midpoint Formula – Explanation and Examples

**The midpoint formula is a method for finding the exact center of a line segment.**

Since a line segment, by definition, is finite, it has two end points. Therefore, another way to think about the midpoint formula is to think of it as a way to find the point exactly between two other points.

The midpoint formula requires us to plot points and a thorough knowledge of fractions.

In this section, we will go over:

**What is the Midpoint Formula?****How to Find the Midpoint of a Line**

## What is the Midpoint Formula?

Given two points (x_{1}, y_{1}) and (x_{2}, y_{2}), the midpoint formula is (^{(x1+x2)}/_{2}, ^{(y1+y2)}/_{2}).

If we are trying to find the center of a line segment, the points (x_{1}, y_{1}) and (x_{2}, y_{2}) are the end points of the line segment.

Notice that the output of the midpoint formula is not a number. It is a set of coordinates, (x, y). That is, the midpoint formula gives us the coordinates for a point that is exactly between the two given points. This is the exact middle of a line segment connecting the two points.

The distance from either point to the midpoint will be exactly half the distance between the two initial points.

## How to Find the Midpoint of a Line

First, pick a point to be (x_{1}, y_{1}) and a point to be (x_{2}, y_{2}). It doesn’t matter much which is which, but in some cases, we may have to determine the coordinates of the two points from a graph.

Then, we can plug the values x_{1}, y_{1}, x_{2}, and y_{2} into the formula (^{(x1+x2)}/_{2}, ^{(y1+y2)}/_{2}).

Remember learning about averages and means? To find the average or mean of two numbers, we add the two numbers together and divide by two. That’s exactly what we are doing in the formula!

Therefore, we can think of the midpoint formula as finding the point that is the average of the x-terms and the y-terms.

## Examples

In this section, we will go over some examples of how to use the midpoint formula and their step-by-step solutions.

### Example 1

Consider a line segment that starts at the origin and ends at the point (0, 4). What is the midpoint of this line?

### Example 1 Solution

It is easy to see that this line is 4 units in length and its midpoint is (2, 0). This makes it easy to illustrate how the midpoint formula works.

First, let’s designate the origin, (0, 0) as (x_{1}, y_{1}) and the point (4, 0) as (x_{2}, y_{2}). Then we can plug them into the midpoint formula:

(^{(x1+x2)}/_{2}, ^{(y1+y2)}/_{2}).

(^{(4+0)}/_{2}, ^{(0+0)}/_{2}).

(^{4}/_{2}, 0)

(2, 0).

This matches with our intuition. After all, the midpoint of 0 and 4 is 2.

### Example 2

Consider a line segment that starts at (0, 2) and ends at (0, 4). What is the midpoint of this line segment?

### Example 2 Solution

Again, we can see that this is a line segment of length 2 units. Its midpoint is one unit from each end point at (0, 3). This once again makes it easy to demonstrate how the midpoint formula works.

Let’s let (0, 2) be (x_{1}, y_{1}) and (0, 4) be (x_{2}, y_{2}). Then, plugging the values into the midpoint formula gives us:

(^{(0+0)}/_{2}, ^{(4+2)}/_{2})

(0, ^{6}/_{2})

(0, 3).

Therefore, the midpoint is (0, 3), and, as before, this matches our intuition.

### Example 3

Find the midpoint of a line segment that extends from (-9, -3) to (18, 2).

### Example 3 Solution

It is not as immediately obvious where the midpoint of this line is. But, we can still assign one point (let’s say (-9, -3) as (x_{1}, y_{1})) and the other point as (x_{2}, y_{2}). Then, we can insert the values into the midnight formula:

(^{(-9+18)}/_{2}, ^{(-3+2)}/_{2})

(^{9}/_{2}, ^{-1}/_{2}).

In this case, we can just leave the two numbers as fractions for our answer. All three points are plotted below.

### Example 4

The graph below features a line segment k. What is the midpoint of the line segment?

### Example 4 Solution

Before we can determine the midpoint of this line segment, we need to find the coordinates of its endpoints. The endpoint in the second quadrant is four units left of the origin and one unit above it. The endpoint in the fourth quadrant is three units to the right of the origin and three units below it. This means that the endpoints are (-4, 1) and (3, -3) respectively. Let’s also have them be (x_{1}, y_{1}) and (x_{2}, y_{2}) respectively.

When we insert these values into the midpoint formula, we get:

(^{(-4+3)}/_{2}, ^{(3+1)}/_{2})

(^{-1}/_{2}, ^{-2}/_{2})

(^{-1}/_{2}, -1).

Therefore, the exact center of this line segment is the point (^{-1}/_{2}, -1).

### Example 5

A scientist finds two nests for an endangered bird on an island. One nest is 1.2 miles north and 1.4 miles east of the scientist’s research facility. The second nest is 2.1 miles south and 0.4 miles east of the facility. The scientist wants to set up one camera in a spot that is as close as possible to both nests in hopes of catching some footage of the birds. Where should she put this camera?

### Example 5 Solution

The spot that will minimize the distance to each nest is the midpoint between the coordinates of the two nests.

Let’s let north and east be the positive directions. Since the first nest is 1.2 miles north and 1.4 miles east, we can plot its coordinates at (1.4, 1.2). Similarly, the coordinates of the second nest are at (0.4, -2.1).

If the coordinates of the first nest are (x_{1}, y_{1}) and the coordinates of the second nest are (x_{2}, y_{2}), then the midpoint is:

(^{(1.4+0.4)}/_{2}, ^{(1.2-2.1)}/_{2})

(^{1.8}/_{2}, ^{-0.9}/_{2})

(0.9, ^{-0.9}/_{2})

That is, the scientist should set up her camera at the coordinates (0.9, ^{-0.9}/_{2}). Since ^{-0.9}/_{2} is -0.45, the camera should be at a spot 0.45 miles north of the facility and 0.9 miles east of it.

### Example 6

The midpoint of a line segment is (9, 4). One of the endpoints of the line segment is (-8, -2). What is the other endpoint of this line segment?

### Example 6 Solution

We can plug the values we know into the midpoint formula and work backwards. We know that the midpoint is (9, 4) and that one end point is (-8, -2). Let’s let this be (x_{1}, y_{1}). Then, we have:

(-8+x_{2})/2=9 and (-2+y_{2})/2=4.

Now, we can multiply both sides of both equations by 2, which gives us:

-8+x_{2}=18 and -2+y_{2}=8.

Finally, adding 8 to both sides of the equation on the left and 2 to both sides of the equation on the right gives us x_{2}=26 and y_{2}=10.

Therefore, the other end point is (26, 10).

### Practice Problems

- A line segment connects the points (9, 1) and (8, 7). What is the midpoint of this line segment?
- A line segment connects the points (-3, -6) and (-7, 1). What is the midpoint of this line segment?
- A line segment connects the points (-105, 207) and (819, 759). What is the midpoint of this line segment?
- An artist plans to create a mural. He plans to paint a star at a point 10 feet to the right of and 5 feet above the bottom left corner of the wall. He also plans to paint a star at the upper left corner. The artist also plans to paint the moon exactly between the two stars. If the wall is 12 feet tall, where should the artist paint the moon?
- A line segment has a midpoint at (-1, -2). If one of the endpoints is (16, 8), what is the other endpoint of the line segment?

### Practice Problems Answer Key

- The midpoint is (
^{17}/_{2}, 4) - This midpoint is (-5,
^{-5}/_{2}) - The midpoint is (357, 483)
- In this case, coordinates of the stars are (10, 5) and (0, 12). The midpoint is (5,
^{17}/_{2}). - The other endpoint is (-18, -12).