Distance and Midpoint Formulas
Coordinate geometry measures segments on a grid.
About this topic
The distance between (x₁, y₁) and (x₂, y₂) is d = √((x₂ − x₁)² + (y₂ − y₁)²), which is the Pythagorean theorem applied to the horizontal and vertical differences. The midpoint is the average of the coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2).
Together these let you prove properties of shapes using coordinates, such as whether a triangle is isosceles. GPS and map software use extended versions of the same idea.
Worked example: find the distance from (1, 2) to (7, 10). The differences are 6 and 8, so d = √(36 + 64) = 10. The midpoint is ((1 + 7)/2, (2 + 10)/2) = (4, 6). If the midpoint of a segment is (3, 5) and one endpoint is (1, 2), the other endpoint is (5, 8), found by doubling the midpoint and subtracting.
Game developers use the distance formula to check whether objects collide, and mapping apps use it for nearby places. A common mistake is subtracting inconsistently, but squaring makes the order harmless as long as both coordinates use the same direction. Another is forgetting the square root at the end, or adding the squares incorrectly with negatives. The midpoint formula averages coordinates, so do not subtract. To show a triangle is isosceles, compute all three side lengths and compare them.
Key ideas
- d = √((x₂−x₁)² + (y₂−y₁)²)
- Midpoint averages the coordinates
- Distance comes from Pythagoras
Quick check
Use cases: where Distance and Midpoint Formulas helps
- Homework: review the key idea behind Distance and Midpoint Formulas before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Distance and Midpoint Formulas without notes.
- Visual learning: play with the interactive lab to see how Distance and Midpoint Formulas changes when you adjust its inputs.
- Teaching a friend: explain Distance and Midpoint Formulas in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Distance and Midpoint Formulas?
Coordinate geometry measures segments on a grid.
What will I learn in Distance and Midpoint Formulas?
d = √((x₂−x₁)² + (y₂−y₁)²). Midpoint averages the coordinates. Distance comes from Pythagoras.
Is Distance and Midpoint Formulas part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Distance and Midpoint Formulas lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Distance and Midpoint Formulas?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.