Midpoint

The midpoint of the segment (x₁, y₁) to (x₂, y₂)

In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints.

In statistics and use of histograms, a midpoint is known as class mark.

Formulas

The formula for determining the midpoint of two points x₁ and x₂ on a line is:

$\frac{x_1 + x_2}{2}$

The formula for determining the midpoint of a segment in the plane, with endpoints (x₁, y₁) and (x₂, y₂) in Cartesian coordinates, is:

$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

The formula for determining the midpoint of a segment in the space, with endpoints (x₁, y₁, z₁) and (x₂, y₂, z₂) is:

$\left(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$

More generally, for an n-dimensional space with axes $x_1, x_2, x_3, \dots, x_n\,\!$, the midpoint of an interval is given by:

$\left(\frac{x_{1_1} + x_{1_2}}{2}, \frac{x_{2_1} + x_{2_2}}{2}, \frac{x_{3_1} + x_{3_2}}{2}, \dots , \frac{x_{n_1} + x_{n_2}}{2} \right)$

Construction

Given two points, finding the midpoint is one of the compass and straightedge constructions. The midpoint of a line segment can be located by first constructing a lens using circular arcs, then connecting the cusps of the lens. The point where the cusp-connecting line intersects the segment is then the midpoint. It is more challenging to locate the midpoint using only a compass, but it is still possible.

Deriving the formula

In order to derive the formula you must understand the relationship between distance, and midpoint. Distance is very important in deriving the formula for midpoint. This is because the two are intertwined because when you have (x₁, y₁) and (x₂, y₂) there is a relationship in distance between these two points. This forms a line segment, but this isn't the relationship we are looking for. The relationship in distance that helps us derive the formula is Pythagorean Theorem. How this helps us is that there is a relationship that (x₁, y₁) and (x₂, y₂) with a third point. This Third point can be achieved by taking either x₁ or x₁ and y₁ or y₁. This third point allows us to make a triangle. which set of points you choose to use depends on which angle you are making the triangle at. Making a triangle helps us by because then we can use A² + B² = C² to find the distance of the long side. The long side of the triangle would be the line segment that we are trying to find the midpoint of. After you use Pythagorean theorem you can find the midpoint of that by dividing it by 2, but that will not give you the coordinate point. This can be helpful because you can then apply the (x₁,x₂) and (y₂,y₁) to a A² or B². This would make the formula C² = (y₂ − y₂)² + (x₁ − x₂)². This formula will not find midpoint, but it will assist someone in deriving the formula for midpoint from a formula that will find distance.

Generalizations

The midpoint is actually an affine invariant. Hence, aforementioned formulas for Cartesian coordinates are feasible for any affine coordinate system.

The midpoint is not defined in projective geometry. Any point inside a projective range may be projectively mapped to any another point inside (the same or some else) projective range. Fixing one of such points as a midpoint actually defines an affine structure on the projective line containing that range. The projective harmonic conjugate of the two endpoints together with the midpoint is the point at infinity.

The definition of the midpoint of a segment may be extended to geodesic arcs on a Riemannian manifold. Note that, unlike affine case, the midpoint between two points may be defined not uniquely.

See also

• Midpoint polygon
• Median (geometry)
• Segment bisector
• Numerical integration

External links

• Animation – showing the characteristics of the midpoint of a line segment