Question

The median of a trapezoid means the line joining the midpoints of the two nonparallel sides. Prove that the median bisects both diagonals; that the median is parallel to the two parallel bases and equal to half the sum of their lengths.

Short answer

The median of a trapezoid bisects the diagonals, is parallel to the bases, and its length is the average of the bases.

Step-by-step solution

Define the Trapezoid and Median

Consider a trapezoid ABCD with bases AB and CD, where AB is parallel to CD. Let E and F be the midpoints of the nonparallel sides AD and BC, respectively. EF is the median of the trapezoid.

Prove the Median Bisects the Diagonals

To prove that the median EF bisects the diagonals AC and BD, observe that triangles ADE and BCF each have a median (EF) connecting the midpoints of two sides. By the midpoint theorem, EF is parallel to AB and CD and half their length. Therefore, EF intersects AC and BD at their midpoints, thus bisecting them.

Prove the Median is Parallel to the Bases

Use the properties of the midline (midpoint theorem), which states that a line segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. Since E and F are midpoints and EF is a median, EF will be parallel to AB and CD.

Prove the Median's Length

According to the properties of a trapezoid, the length of the median EF can be found as the average of the lengths of the two bases AB and CD. Hence, EF = (AB + CD) / 2.

Key concepts

Trapezoid Median

A trapezoid is a quadrilateral with at least one pair of parallel sides. These parallel sides are called the bases. The nonparallel sides are referred to as the legs.
The median of a trapezoid is the line segment that connects the midpoints of the two nonparallel sides. If you have a trapezoid ABCD with bases AB and CD, let’s denote E and F as the midpoints of the legs AD and BC, respectively. The segment EF is what we refer to as the median of the trapezoid. It holds special geometric properties that help us understand the shape better.
Understanding the median is crucial for various proofs and applications, as it often simplifies complex problems involving trapezoids. The median is uniquely significant because it divides the trapezoid into two smaller trapezoids of equal area.

Midline Theorem

The midline theorem (or midpoint theorem) states that a line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half as long.
This theorem is essential in understanding the properties of the trapezoid’s median. In the context of our trapezoid, we can create two smaller triangles, ADE and BCF, where E and F are the midpoints of AD and BC.
Applying the midline theorem in trapezoid ABCD:

• In triangle ADE, EF is parallel to DE and is half its length.
• In triangle BCF, EF is parallel to CF and is half its length.

Therefore, by the midline theorem, EF is parallel to both AB and CD, and this first step helps us in proving other properties, like the median’s length and its relation to the diagonals.

Diagonal Bisection

One fascinating property of the median in a trapezoid is that it bisects the diagonals. This means that the median cuts the diagonals into equal segments.
Let's demonstrate how this works. Consider the diagonals AC and BD in trapezoid ABCD, and let EF be the median. By constructing the triangles ADE and BCF, where E and F are the midpoints of AD and BC, the median EF shows some interesting qualities:

• Triangle ADE with median EF shows that EF is parallel to AB and extends to intersect AC at its midpoint.
• Similarly, in triangle BCF, the median EF is parallel to CD and intersects BD at its midpoint.

Hence, EF bisects the diagonals. This property is vital because it simplifies many complex geometric problems and proofs. It ensures that any segment of the diagonals intersected by the median is always equal, simplifying calculations and constructions within trapezoids.