Two formulas cover most coordinate geometry questions: how far apart two points are, and the point exactly halfway between them.
The distance formula
For points (x₁, y₁) and (x₂, y₂):
d = √((x₂ − x₁)² + (y₂ − y₁)²)
It’s the Pythagorean theorem in disguise. The horizontal change and the vertical change are the two legs of a right triangle, and the distance is the hypotenuse.
Example Find the distance between (1, 1) and (5, 4)
- Horizontal change: 5 − 1 = 4. Vertical change: 4 − 1 = 3.
- d = √(4² + 3²) = √25 = 5.
Example Find the distance between (−2, 3) and (4, −5)
- Changes: 4 − (−2) = 6 and −5 − 3 = −8.
- d = √(36 + 64) = √100 = 10.
The order of the points doesn’t matter, because squaring makes every change positive.
Example Find the distance between (0, 2) and (3, 7)
- d = √(3² + 5²) = √34.
- √34 doesn’t simplify, so leave it, or round: about 5.83.
Subtracting a negative
From −2 to 4 is 4 − (−2) = 6, not 2. Write the parentheses every time a coordinate is negative.
Taking the root too soon
√(36 + 64) is 10. √36 + √64 is 14. Add the squares first, then take the root.
The midpoint formula
The midpoint averages the x-coordinates and averages the y-coordinates:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Example Find the midpoint of (2, 7) and (8, −1)
M = ((2 + 8)/2, (7 + (−1))/2) = (5, 3).
Example Find the midpoint of (−3, 4) and (2, 9)
M = (−1/2, 13/2), or (−0.5, 6.5).
Finding an endpoint from the midpoint
If you know one endpoint and the midpoint, the midpoint is halfway, so take the same step again.
Example M(3, −2) is the midpoint of AB. A = (1, 4). Find B.
- From A to M, x goes up 2 and y goes down 6.
- Do it again from M: B = (3 + 2, −2 − 6) = (5, −8).
The algebra version: (1 + x)/2 = 3 gives x = 5, and (4 + y)/2 = −2 gives y = −8.
Putting them to work
These two formulas answer a lot of shape questions:
- Is a triangle isosceles? Find all three side lengths and look for two equal ones.
- Is a triangle right? Check whether the side lengths satisfy a² + b² = c².
- Is a quadrilateral a parallelogram? Its diagonals bisect each other, so they share a midpoint.
- Circle equations. The radius is the distance from the center to a point on the circle. The circles guide shows how to write the equation.
Example Is the triangle with vertices (0, 0), (4, 0) and (2, 5) isosceles?
- From (0, 0) to (2, 5): √(4 + 25) = √29.
- From (4, 0) to (2, 5): √(4 + 25) = √29.
- Two sides are equal, so yes.
In three dimensions
Add a third term: d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²). The distance from (1, 2, 3) to (3, 5, 9) is √(4 + 9 + 36) = 7.
Your turn
Find the distance between (3, 2) and (9, 10).
Show answer
10. √(36 + 64).
Find the distance between (−1, −4) and (4, 8).
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13. √(25 + 144).
Find the distance between (2, 1) and (5, 3).
Show answer
√13, about 3.61.
Find the midpoint of (−6, 2) and (4, 8).
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(−1, 5).
M(4, 1) is the midpoint of PQ, and P = (−2, 5). Find Q.
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(10, −3).
A circle has center (1, 2) and passes through (4, 6). Find the radius.
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5.