## Splitting a Square

What do you notice in the story below? What are you wondering about? Leave a comment to tell us your thoughts!

A square is drawn on a coordinate plane. The square is in the first quadrant and the sides of the square are parallel to the axes.

A line is drawn from the origin that cuts the square into two sections whose areas are in a ratio of 2:1.

What a great question starter! It links geometry and proportions and area and maybe more .

Is there any feedback about the problems students solve and the strategies they use?

Hi,

Thanks for the interesting puzzle. Equally interesting are the reflections about how to approach the problem. Consider the statement of the problem. What is the easiest case to visualize? For me it was the square with one corner at the origin. Generalize. But is the problem limited to that case? No. It must only be a square oriented to the axes and in the first quadrant. Classification. How to characterize the different cases of location of the square? I chose corner closest to the originand further split the squares into those where that corner was at the origin, was on the line y = x, was on an axis, or was none of the other cases. Visualize the shapes created by the line. Solutions vary based on classification.

Solution for corner at the origin. Look for integer area solutions. Consider n squared. The only way it can be divisible by 3 is if n is divisible by 3. The area of a triangle = 1/2bh so n should also be divisible by 2. So choose n = 6, n squared = 36, A triangle = 1/2*6h = 36/3,

or h= 4. Proportions. Then the line must pass from the origin to a point on a vertical side 4/6 = 2/3 of the way from the x axis. Proof that this is the general solution for corner at the origin would involve formulas and proportions showing that the areas for the specified triangle and square would both increase with n squared.

Hope this helps!

A square can be of any size and be located anywhere in the quadrant. That is to say there are an infinite number of possibilities describing such a square. For each possibility there will be two lines that cut the square into 2:1 proportions. The first will leave the larger area above the smaller; the second will leave the smaller area over the larger. More simply, the two lines will split the square into equal thirds.

Now the challenge is to determine how to split a square into thirds, and then try to relate the line equations to the coordinates of the square!

This is my first visit to your site and it has been very enjoyable ;-)