# 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. This theorem is a fundamental concept in geometry and has various applications in mathematics, particularly in the study of quadrilaterals.

## 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
### Construction proof

The simplest proof that one can obtain is done by some extra construction.
Let ABCD be a trapezoid ($AD\parallel BC,AD>BC$). Construct $E$ on $AD$ such that $BC=ED$ (and, therefore, $BCDE$ is a parallelogram). Let $M,P,N$ be the midpoints of $AB,BE$ and $CD$ respectively.
Since $BCDE$ is a parallelogram, $PN=BC$. Using the midpoint theorem,
    $MP={\frac {1}{2}}AE.$
Since $AE=AD-BC,$
    $MP={\frac {AD-BC}{2}}.$
Summing this with $PN$, we get
    $MN={\frac {AD+BC}{2}},$
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

The Median of the Trapezoid theorem is widely used in solving geometric problems involving trapezoids. It simplifies the calculations of distances and areas in problems where trapezoids are involved.

### 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. The Median of the Trapezoid theorem, in particular, exemplifies the elegance and simplicity that can be found in geometric theorems, making it a notable topic of study in mathematical education and research.

## Related concepts

* Trapezoid
* Midsegment theorem
* Quadrilaterals