Midpoint & Distance Formulas

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Midpoint Formula

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

The midpoint is the middle point of a line segment. It is equidistant from both endpoints. The formula for determining the midpoint of a segment in the plane, with endpoints \small (x_1, y_1) and \small (x_2, y_2) is

M=\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).

In three-dimensional Cartesian space, the midpoint formula is

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

Distance Formula

In Geometry, the distance between two points of the xy-plane can be found using the distance formula. The distance between (x1, y1) and (x2, y2) is given by

d=\sqrt{(\Delta x)^2+(\Delta y)^2}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.\,

Similarly, given points (x1, y1, z1) and (x2, y2, z2) in three-space (xyz-coordinates), the distance between them is

d=\sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}.

Which is easily proven by constructing a right triangle with a leg on the hypotenuse of another (with the other leg orthogonal to the plane that contains the 1st triangle) and applying the Pythagorean Theorem.

In the study of complicated geometries, we call this (most common) type of distance Euclidean distance, as it is derived from the Pythagorean Theorem, which does not hold in Non-Euclidean geometries. This distance formula can also be expanded into the arc-length formula.

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