The distance formula is Pythagoras in disguise.
d = √(x₂ − x₁)² + (y₂ − y₁)²
There is no new formula to learn. The two coordinate gaps are the legs; the distance is the hypotenuse. a² + b² = c², written with subtractions.
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- Distance and the Midpoint of a Line Segment
Distance and the Midpoint of a Line Segment
Distance squares and adds. Midpoint just averages.
Find the distance
How far apart are A(2, 3) and B(8, 11)?
The horizontal gap is 8 − 2 = 6 and the vertical gap is 11 − 3 = 8. These are the legs of a right triangle, so the distance is √36 + 64 = √100 = 10. D adds the two gaps, which would be the path along the grid, not the straight line. A and B give only one gap each — the distance must be longer than either.
Worked through, with the order not mattering.
A(2, 3), B(8, 11)
the two points
(8 − 2)² + (11 − 3)²
= 36 + 64
d = √100 = 10
the answer
Subtract in either order — but square before adding
(2 − 8)² is also 36, so the order of the points does not matter. What does matter is squaring each gap before adding: √36 + 64 is 10, but √36 + √64 is 14.
Midpoint: average each coordinate.
M = (x₁ + x₂2, y₁ + y₂2)
A(2, 3), B(8, 11)
same two points
x: 2 + 82 = 5
halfway across
y: 3 + 112 = 7
halfway up
M(5, 7)
the midpoint
Sanity check: 5 sits between 2 and 8, and 7 between 3 and 11. A midpoint always lands between the two endpoints, in both coordinates.
Add for the midpoint, subtract for the distance
Swapping these is the classic error. A midpoint is an average, so it uses a plus. A distance is a gap, so it uses a minus.
Negative coordinates: let the brackets do the work.
P(−3, 5), Q(1, −2)
find the distance
(1 − (−3))² = 4² = 16
subtracting a negative
(−2 − 5)² = (−7)² = 49
squaring kills the sign
d = √65 ≈ 8.06
leave it exact if asked
A distance is never negative, so if your final answer is, a sign went astray before the squaring.
Working backwards to a missing endpoint.
M(5, 7) is the midpoint of A(2, 3) and B
find B
2 + x2 = 5 → x = 8
solve for the x-coordinate
3 + y2 = 7 → y = 11
and the y-coordinate
B(8, 11)
matching step 3 ✓
Faster route
A to M is +3 across, +4 up
so M to B is the same again
Apply it
(5 + 3, 7 + 4)
= (8, 11) ✓
A missing coordinate from a distance, and the same-x shortcut.
Given the distance, the unknown gap comes from a² + b² = c² — and it can go either way.
The point (2, y) is 13 units from (−3, 1). Find y.
gap across = 2 − (−3) = 5
the known leg
5² + g² = 13²
g is the vertical gap
g² = 169 − 25 = 144, g = 12
the missing leg
y = 1 + 12 = 13 or y = 1 − 12 = −11
up or down from 1
Two answers, not one
A gap of 12 can be 12 up or 12 down, so both (2, 13) and (2, −11) are 13 units from (−3, 1). Giving only y = 13 finds half the answer; a multiple-choice item may list either one.
Find the distance between (−4, −6) and (−4, 3).
same x, so the segment is vertical
no across gap
d = 3 − (−6) = 9
top minus bottom
The formula still works — √0² + 9² = 9 — but when one coordinate matches, the distance is just the difference of the other, made positive.
Variations: an isosceles test, fraction midpoints, the origin as midpoint.
The same two formulas, used for something other than a single answer.
Is the triangle A(0, 0), B(4, 2), C(2, 4) isosceles?
AB = √4² + 2² = √20
gaps 4 and 2
AC = √2² + 4² = √20
gaps 2 and 4
BC = √2² + 2² = √8
gaps 2 and 2
AB = AC, so yes. To compare lengths we never need the decimal: compare what is under the root.
Midpoint of (3, −1) and (−4, 6)
x: 3 + (−4)2 = −12
an odd sum
y: −1 + 62 = 52
M(−12, 52)
leave the halves
When a sum is odd the midpoint lands between grid points. That is correct, not a slip.
The midpoint of A(7, −2) and B is the origin. Find B.
7 + x2 = 0 → x = −7
−2 + y2 = 0 → y = 2
B(−7, 2)
every sign of A flipped
When the midpoint is (0, 0), the other endpoint is simply A with both signs changed.
On the exam
Exams ask for a distance, a midpoint, a missing endpoint or a missing coordinate; the harder items ask for a triangle's perimeter or whether it is isosceles or right. The fastest route is to write the across and up gaps first, then use Pythagoras for distance and averages for the midpoint. For a 10-second check, the midpoint must sit between the endpoints, and a distance must be longer than either gap but shorter than their sum.
Now you try
Practice problem
Find the midpoint
What is the midpoint of (−6, 4) and (2, −10)?
Average each coordinate: −6 + 22 = −2 and 4 + (−10)2 = −3, so the midpoint is (−2, −3). Check it lies between: −2 is between −6 and 2 ✓, and −3 is between 4 and −10 ✓ C subtracts instead of averaging, which is the distance calculation, not the midpoint.
Hint: a midpoint always sits between the two endpoints. Check that before choosing — it eliminates two options straight away.