What is a midpoint?
The midpoint of a line segment is the point exactly halfway between its two endpoints. It splits the segment into two equal halves, so it is the same distance from both ends. Its x is the average of the two x values, and its y is the average of the two y values.
The center of a circle is the midpoint of any diameter: a diameter from (−1, −4) to (5, −4) has its center at (2, −4).
Source OpenStax · College Algebra 2e, 2.1 The Rectangular Coordinate Systems and Graphs ↗
How to find the midpoint
Here it is with your numbers:
- x = (7 + 9) ÷ 2
- x = 8
- y = (−2 + 5) ÷ 2
- y = 1.5
It means (8, 1.5) is the same distance, 3.6401, from both ends.
To find a missing endpoint, turn the formula around: double the midpoint and take away the end you know, once for x and once for y. Choose “Endpoint” above to do it here.
Length of the segment
The tiles also show how long the segment is, from the distance formula d = √((x₂ − x₁)² + (y₂ − y₁)²), which comes from the Pythagorean theorem. From (−3, −1) to (2, 3) is √(5² + 4²), which is √41, or 6.4031. Each half is half of that, and the slope is the rise over the run.
Source Common Core State Standards · Grade 8, distance between two points (8.G.B.8) ↗
Midpoint examples
| Between | Midpoint |
|---|---|
| (7, −2) and (9, 5) | (8, 1.5) |
| (−1, −4) and (5, −4) | (2, −4) |
| (0, 0) and (10, 6) | (5, 3) |
| 70 and 90 | 80 |
| 24 and 40 | 32 |
FAQs
What is the midpoint formula?
For endpoints (x₁, y₁) and (x₂, y₂), the midpoint is ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2): the average of the x values and the average of the y values. The midpoint of (7, −2) and (9, 5) is (8, 1.5).
Source OpenStax · College Algebra 2e, 2.1 The Rectangular Coordinate Systems and Graphs ↗
How do you find the middle point between two numbers?
Add the two numbers and divide by 2. It is the same as the midpoint formula with only x values: halfway between 3 and 11 is (3 + 11) ÷ 2, which is 7.
What is the midpoint of 70 and 90?
80. 70 + 90 is 160, and 160 ÷ 2 is 80, which is 10 away from both numbers.
What is the midpoint of 24 and 40?
32. 24 + 40 is 64, and 64 ÷ 2 is 32, which is 8 away from both numbers.
What is the formula for end point?
Turn the midpoint formula around: the other endpoint is (2 × xₘ − x₁, 2 × yₘ − y₁), where (xₘ, yₘ) is the midpoint and (x₁, y₁) is the end you know. If one end is (−1, −4) and the midpoint is (2, −4), the other end is (5, −4).
Sources
- OpenStax: College Algebra 2e, 2.1 The Rectangular Coordinate Systems and Graphs
- Common Core State Standards: Grade 8, distance between two points (8.G.B.8)