Varignon's Theorem
Varignon's Theorem
Varignon's Theorem states that the midpoints of any quadrilateral form a parallelogram.
Parallelograms
So what is a parallelogram? There are four criteria for a shape to be classified as a parallelogram.
The first criteria is that the opposite angles must be congruent. Consider the following quadrilateral. What can we change to make this a parallelogram?
The second criteria for a parallelogram is that the opposite sides must be the same length. Consider the next quadrilateral. What can we move to make this a parallelogram?
The third criteria for a parallelogram is that the opposite sides are parallel. Consider the following quadrilateral. What can we move to make this a parallelogram? (Hint: two lines are parallel if they have the same slope.)
The fourth criteria for a parallelogram is for the diagonals to bisect each other, or cut each other perfectly in half. Consider this quadrilateral. What can we change to turn this into a parallelogram?
Now that we know what a parallelogram is, we can see that Varignon's Theorem is true. Play around with the following diagram to watch the parallelogram change with the quadrilateral. What shape do you get when both diagonals are the same length? What shape do you get when the diagonals are perpendicular to each other?