Earlier today as I peddled my bicycle down to the waterfront to supervise my 10-year old and a few of his friends, I stopped dead in my (tire) tracks because I saw this:

To the untrained eye, that may look like just a lot of grass, a soccer goal, some trees, and a breathtakingly blue sky. Zoom in and you might see two figures, one at a far corner of the soccer field and one in the center. What were these two figures doing? They were lining the soccer field for the first time this summer. But I knew what they were really doing. They were doing math.

Me: Hey!

Them: Hey!

Me: Are you using the Pythagorean Theorem?

Them: Yes!

What you can’t see in the first picture (and can see better here) is that they had a 300-foot tape measure stretching across the diagonal of the rectangle. The harsh New Hampshire winter caused one of the set four corners to disappear so they were trying to find the place where it should have been so they could paint the lines. It’s not a FIFA-World-Cup regulation field, but it should be a rectangle anyway. When you are painting lines, it can be deceiving and you can easily paint a quadrilateral but not a rectangle.

Here are three quadrilaterals to illustrate my point. The “x” represents the missing corner. In the first two cases, the missing corner is too far away or too close to the other corners. It distorts the rectangle so that there is only one right angle (lower-left). In the third case, the missing corner is just right and this makes all of the angles ninety degrees.

So, what the Recreation Director was doing was checking to see, via the Pythagorean Theorem, if had found the correct location of the missing corner. Everyone remembers the Pythagorean Theorem! *The sum of the squares of the legs of ANY right triangle must equal the square of the hypotenuse.* You know,

He knew his distance from lower left to upper left was 180 feet and his distance from lower left to lower right was 300 feet since those three corners were still there. This meant that the diagonal across the rectangle, *c*, from lower left to upper right would have to satisfy the equation:

When you solve this quadratic, *c* should be approximately 350. His measuring tape only went to 300 feet, so that presented some more challenges and I am not sure how he dealt with those. He was still working on the field when I pedaled back to my cabin a few hours later and he told me, “My math teacher would be so proud!” I replied, “Not just your math teacher, ALL math teachers! I’m going to write a blog post about this!”

And so I did.

If you’d like to read another post from the “When Are We Ever Going to Use This?” File, click here.

If you’d like a mathematical circuit for Pythagorean Theorem, click here.

Setting: Wolfeboro Camp School