# Median of the trapezoid theorem

The Median of the Trapezoid theorem states that the median of a trapezoid is equal in length to the average of the lengths of the two bases.
In elementary geometry, it is a theorem that the median of a trapezoid is equal in length to the average of the lengths of the two bases. Here, a trapezoid is a quadrilateral with two opposite sides parallel, and its median is defined as the line segment connecting the midpoints of the two non-parallel sides of the trapezoid. The median is parallel to the two parallel sides, and equidistant from them.
One application of the theorem is to simplify the statement of the area formula for a trapezoid. The area is the product of the median length and the height of the trapezoid (the distance between its two parallel sides).

## Statement

Given a trapezoid ABCD, with bases AB and CD, the median (or midsegment) EF is the segment that joins the midpoints of the non-parallel sides AD and BC. The theorem asserts that the length of the median EF is equal to the average of the lengths of the two bases AB and CD. Mathematically, this is expressed as:
$EF={\frac {AB+CD}{2}}$.

## Proof
### Proof as corollary of midpoint theorem

Let $ABCD$ be a trapezoid
Let $E$ and $G$ be the midpoints of $AD$ and $BC$.
Let $F$ be the midpoint of diagonal $DB$
To prove:
* $EG\parallel AB$
* $EG\parallel DC$
* $EG={\frac {1}{2}}(AB+DC)$
Proof:
In $\triangle ADB$, construct $EF$
By the triangular midpoint theorem, $EF\parallel AB$ and $EF={\frac {1}{2}}AB$
In $\triangle DBC$, construct $FG$
By the triangular midpoint theorem, $FG\parallel DC$ and $FG={\frac {1}{2}}DC$
By definition of trapezoid, $AB\parallel DC$
since $FG\parallel DC$, by transitivity $FG\parallel AB$
By parallel postulate, only 1 line exists through $F$ where $FG\parallel AB$, therefore $FG$ is colinear with $EF$
Therefore
$EG\parallel AB$
$EG\parallel DC$
$EG=EF+FG={\frac {1}{2}}(AB+DC)$
Q.E.D.

### Coordinate Geometry Proof

Consider trapezoid ABCD in a coordinate plane, with vertices A(x₁, y₁), B(x₂, y₁), C(x₃, y₂), and D(x₄, y₂), where AB is parallel to CD. The midpoints of AD and BC are given by: $E=\left({\frac {x_{1}+x_{4}}{2}},{\frac {y_{1}+y_{2}}{2}}\right)$ and $F=\left({\frac {x_{2}+x_{3}}{2}},{\frac {y_{1}+y_{2}}{2}}\right)$
The length of the median EF is then calculated as: $EF={\sqrt {\left({\frac {x_{2}+x_{3}}{2}}-{\frac {x_{1}+x_{4}}{2}}\right)^{2}+\left({\frac {y_{1}+y_{2}}{2}}-{\frac {y_{1}+y_{2}}{2}}\right)^{2}}}={\frac {|x_{2}-x_{1}+x_{3}-x_{4}|}{2}}$
Since AB and CD are parallel, the distance between the x-coordinates of AB and CD is constant, and the length of EF is the average of AB and CD.

## Applications
### Area of a trapezoid

One direct application is in deriving the formula for the area of a trapezoid. The area can be calculated as the product of the median and the height (the perpendicular distance between the bases): ${\text{Area}}=EF\times {\text{Height}}$

## Historical context

The study of trapezoids and their properties has a long history in geometry, with the Median of the Trapezoid theorem being one of the key results. The theorem, which states that the median of a trapezoid is equal to the average of the lengths of the two bases, plays a fundamental role in understanding the geometry of trapezoids. Historically, trapezoids and their properties have been studied by mathematicians for centuries, with early references to the shapes and their characteristics found in ancient Greek mathematics. The Median of the Trapezoid theorem itself is an example of how simple geometric principles can yield powerful results, and it has been utilized in various mathematical applications and proofs throughout history. According to Howard Eves, a prominent historian of mathematics, the study of quadrilaterals like the trapezoid has contributed significantly to the development of geometry as a whole.