Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Bimedians are neat characteristics of a quadrilateral. This lesson will define bimedians of a quadrilateral and show (with proof) how they bisect each other.

Suppose you are creating a garden on a small lot of land that is four-sided; call its vertices A, B, C, and D. You want to plant azaleas, begonias, bluebells, and buttercups. To do this, you need to divide the lot into four parts using two boards that intersect the sides of the lot at their midpoints; call the midpoints E, F, G, and H. This is all shown in the image.

The lot of land is a quadrilateral, or a four-sided and two-dimensional shape. The boards are special parts of the quadrilateral lot of land called bimedians, where bimedians are line segments that connect the midpoints of opposite sides of a quadrilateral. Furthermore, the point at which the two boards, or bimedians, intersect (point J) is called the centroid of the vertices of the quadrilateral lot of land.

Now, suppose you know that the boards, or bimedians, have the following lengths:

• EG = 8 feet
• HF = 14 feet

To create your garden, you need to know the lengths of each part of these bimedians that their intersection point divides them into. That is, we need to find lengths EJ, JG, HJ, and JF.

Thankfully, bimedians of a quadrilateral satisfy a certain property that will make finding these lengths quite simple!

## Bisecting Property of Bimedians

As it turns out, the bimedians of a quadrilateral bisect each other. This means that the point of intersection of the bimedians, or the centroid of the vertices of the quadrilateral, cuts each of the bimedians exactly in half.

Ah-ha! This makes finding the lengths in our example a cinch! We simply divide the length of the bimedians in half (or divide by 2).

• 8 ÷ 2 = 4
• 14 ÷ 2 = 7

This tells us that EJ and JG have lengths 4 feet, and HJ and JF have lengths 7 feet. You're all set! You can create your garden now! However, this gets us to wondering: how do we know for sure that the bimedians of a quadrilateral bisect each other? Let's consider this!

## Proof that the Bimedians of a Quadrilateral Bisect Each Other

Thankfully, we have a couple of well-known theorems that make the proof of bimedians bisecting each other pretty simple, and those two theorems are as follows:

1. The diagonals of any parallelogram bisect each other, where a parallelogram is a quadrilateral in which opposite sides are parallel and have equal length.
2. Varignon's theorem - The midpoints of the sides of a quadrilateral form the vertices of a parallelogram.

Those theorems seem pretty straightforward. Let's see how we can use them to easily prove that the bimedians of a quadrilateral bisect each other.

Proof:

Let ABCD be an arbitrary quadrilateral, and let E be the midpoint of the side AB, F be the midpoint of side BC, G be the midpoint of side CD, and H be the midpoint of side DA, where AB and CD are opposite sides and BC and DA are opposite sides. Therefore, the bimedians of quadrilateral ABCD are EG and FH.

By Varignon's theorem, the midpoints form the parallelogram EFGH, and this parallelogram has diagonals EG and FH, and EG and FH are also the bimedians of quadrilateral ABCD.

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