Same Math › Geometry · grades 9–10 › Distance and midpoint formulas
How do the distance and midpoint formulas work?
The midpoint is the average of the coordinates: add the x's and halve, add the y's and halve. The distance between two points is the Pythagorean theorem in disguise: the horizontal change and the vertical change are the legs of a right triangle, so d = √((x₂ − x₁)² + (y₂ − y₁)²). Drawing the triangle makes the formula make sense.
- Legs: the change in x and the change in y.
- Distance: the hypotenuse, by Pythagoras.
- Midpoint: average the x's, average the y's.
Why schools teach it
Geometry on a grid
Eighth graders find distances on a grid with the Pythagorean theorem (8.G.B.8). In Geometry, kids use coordinates to find perimeters and areas (HSG.GPE.B.7), split segments (HSG.GPE.B.6) and prove things like whether a shape is a parallelogram.
Kids who draw the right triangle rarely need to memorize the formula, and they catch sign mistakes because lengths are never negative.
At the kitchen table
What to say, and what to skip
Say this
How far across, and how far up?
Those are the legs.What's the theorem for the hypotenuse?
a² + b² = c².For the midpoint, what's halfway between the x's?
Then the y's.Does the midpoint look halfway on the graph?
Skip this
- Subtracting for the midpoint.Midpoint adds and halves; distance subtracts.
- Sign slips with negatives.5 − (−3) = 8, not 2.
- Forgetting the square root at the end.√100 = 10, not 100.
See it in action
Points (−1, 2) and (5, 10)
Midpoint: average each coordinate
((−1 + 5)/2, (2 + 10)/2) = (2, 6)Distance: the legs
5 − (−1) = 6, 10 − 2 = 8Pythagoras
√(36 + 64) = 10
If the sum under the root isn't a perfect square, simplify the radical: √72 = 6√2.
Free practice sheet
Make a distance and midpoint sheet
Loading practice sheet…
Fridge cheat sheets
Math Fridge Cheat Sheets
Every K–8 math method in one printable book (159 pages): a cheat sheet page like this guide, plus a practice sheet and its answer key for each, in Classic and handwritten Notebook styles. $9 one time, or all 7 subjects for $29.99. See all prices.
Teacher? This method has a classroom pack on TeachersPayTeachers: an anchor chart, a parent letter and practice sheets with answer keys, $4. See the pack
Parents: tell your child's teacher about this guide
Parents also ask
Does the order of the points matter?
No. The differences get squared, so the signs disappear.
What if I know the midpoint and one endpoint?
Work backwards: the endpoint is as far past the midpoint as the start is before it.
Why not just count squares?
That works for horizontal and vertical segments. Slanted ones need Pythagoras.
Is there a 3D version?
Yes: add a (z₂ − z₁)² under the root.