How to Draw a Parallelogram with One Ruler

Posted: March 8, 2018 in Mathematics

Usually in school, you learn how to construct a parallelogram with compass and straightedge. But today, I am gonna show you how to construct a parallelogram with only one ruler. Of course, you need a piece of paper, a pencil, a hand, and your brain as well, etc …

Actually the method is very simple, even an eight year old can do it I promise. So here it is:

  1. Draw a quadrilateral (4-sided shape), any quadrilateral will do even the non-convex one.
  2. Find the midpoint of all four sides with your ruler.
  3. Connect those four midpoints together and you are done.

Here are some examples.

DrawingParallelogram

The proof is very simple.

Let the four points A(a_1, a_2)B(b_1, b_2)C(c_1, c_2)D(d_1, d_2) be the vertices of a quadrilateral.

DrawingParallelogramProof

Then the four midpoints are

\displaystyle P\left(\frac{a_1+b_1}{2}, \frac{a_2+b_2}{2}\right), Q\left(\frac{b_1+c_1}{2}, \frac{b_2+c_2}{2}\right), R\left(\frac{c_1+d_1}{2}, \frac{c_2+d_2}{2}\right), S\left(\frac{d_1+a_1}{2}, \frac{d_2+a_2}{2}\right)

Find the slopes of the sides and show that the opposite sides are parallel:

m_{PQ} = \dfrac{c_2-a_2}{c_1-a_1} = m_{RS} \Rightarrow PQ \;//\; RS

m_{QR} = \dfrac{d_2-b_2}{d_1-b_1} = m_{SP} \Rightarrow QR \;//\; SP

Therefore PQRS is a parallelogram.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

w

Connecting to %s