Library / Geometry, coordinates, surveying and trigonometry / GEO-29
Geometry, coordinates, surveying and trigonometry
Perpendicular bisector of two points
The constant C of the line Ax + By = C that is equidistant from two points, with A = 2(x₂ − x₁) and B = 2(y₂ − y₁).
- Ref
- GEO-29
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- C [cm²]
Every point on the perpendicular bisector of two points is equally far from both. Setting the two distances equal and cancelling squares gives the line Ax + By = C with A = 2(x₂ − x₁), B = 2(y₂ − y₁) and C = x₂² + y₂² − x₁² − y₁². This value of C is shown; A and B are in the derived rows. The line passes through the midpoint, at a right angle to the segment.
The constant C of the line Ax + By = C that is equidistant from two points, with A = 2(x₂ − x₁) and B = 2(y₂ − y₁). A and B are in the derived rows. The two points must be different.
Calculator
- Coefficient A, 2(x₂ − x₁)8 cm
- Coefficient B, 2(y₂ − y₁)4 cm
- Distance between the points4.472 cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- For (1, 1) and (5, 3) the bisector is 8x + 4y = 32; the midpoint (3, 2) satisfies it.
- Two bisectors of the sides of a triangle meet at its circumcentre, the centre of the circle through the three points.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| x1 Point 1, x | −10 to 10 cm | −67 | 33 | 100 | +0.4 cm | −3 % |
| y1 Point 1, y | −10 to 10 cm | −67 | 33 | 100 | +0.4 cm | −3 % |
| x2 Point 2, x | −10 to 10 cm | 7 | 107 | 100 | +0.4 cm | +13 % |
| y2 Point 2, y | −10 to 10 cm | 23 | 123 | 100 | +0.4 cm | +8 % |
Sweep of Point 2, x
| Point 2, x (cm) | Line constant C (cm²) |
|---|---|
| −10 | 107 |
| −7.5 | 63.25 |
| −5 | 32 |
| −2.5 | 13.25 |
| 0 | 7 |
| 2.5 | 13.25 |
| 5 | 32 |
| 7.5 | 63.25 |
| 10 | 107 |
Sweep one variable
Variables and how each one contributes
Line constant C
What the equation gives you. Everything else feeds it.
Point 1, x
The x-coordinate of the first point, in cm. Lowers C and also tilts the line. Raising it by 0.5 (to 1.5) moves the result by −3.91 %, from 32 to 30.75 cm².
Point 1, y
The y-coordinate of the first point, in cm. Lowers C and also tilts the line. Raising it by 0.5 (to 1.5) moves the result by −3.91 %, from 32 to 30.75 cm².
Point 2, x
The x-coordinate of the second point, in cm. Raises C and also tilts the line. Raising it by 0.5 (to 5.5) moves the result by +16.4 %, from 32 to 37.25 cm².
Point 2, y
The y-coordinate of the second point, in cm. Raises C and also tilts the line. Raising it by 0.5 (to 3.5) moves the result by +10.2 %, from 32 to 35.25 cm².
Worked example with the values above
- A = 2(5 − 1) = 8 and B = 2(3 − 1) = 4.
- Squares: 34 − 2.
- C = 32 cm², so the line is 8x + 4y = 32.
Assumptions and limits
- Two distinct points in one Cartesian frame and one unit.
- The line is written as Ax + By = C, with the coefficients taken exactly as stated.
Common mistakes
- Writing the line through the midpoint with the slope of the segment instead of the perpendicular slope.
- Dropping the factor 2 from A and B but not from C.
- Using the points as the same point, which has no unique bisector.
Statistics and uncertainty
These are exact algebraic results from the coordinates, so the uncertainty comes from the inputs. Near-parallel lines and nearly collinear points are the cases where small input errors give large changes in angles or areas.
Where it is used
- Clearances, offsets and angles between members in drawings, CAD and surveying.
- Areas of plots, plates and polygon pieces from corner coordinates.
Maths you need
- Gradient: rise over run of a line
- Vector: direction and length written as (x, y)
- Dot product: the sum of products of components