## Mathochism: The power of triangles

**One woman’s attempt to revisit the math that plagued her in school. But can determination make up for 25 years of math neglect?
**

In the last two classes, Uchitel has been making a concerted effort to catch up. “Speed geometry,” he calls it, and fortunately, the concepts involved are not that tricky, or else I and the rest of the class would be completely lost.

We’ve covered two chapters in two days, and because of that, our next test won’t just cover chapters 3 and 4, but the end of chapter 2 and all of chapter 5. This hasn’t quite started freaking me out yet, but I could tell my classmates were starting to feel panic. There was a lot of muttering, and a lot of “what is going to be on the test” by the end of class.

Uchitel was trying his best to soothe, promising a study guide and a review session Tuesday. Oh, and extra credit for flash cards. While I appreciate these gestures, I worry a little. It’s not that I think he will go back on his word — it’s more that the mathochist in me doesn’t want things too easy. I still have pre-calc and calculus to go, and I need someone like the Dour Professor to push me along.

Then again, the Brofessor’s tests were impossible, and his teaching lackluster. So too many challenges are not good either.

I am learning, though. And proofs still don’t feel natural. But I believe I have discovered the linchpin of geometry; the knowledge I must keep in my brain files so I can continue this journey. It’s all about triangles.

Triangles, Uchitel told us, are key. Sure, trapezoids and kites are fun. But triangles will follow us forever, mainly because their shape and their angles apply to so many things, including measuring distances and figuring out trajectories. I was just reading about a probe that NASA sent to Mercury. It took that probe six years to get there, which confused me. Surely Mercury is far closer to Earth than Jupiter?

But as it turns out, the reason it took so long was that Mercury is too close to the Sun, and, if the probe was going too fast, the Sun’s gravitational field could easily have sucked the probe in. So the engineers had to calculate a trajectory (using angles!) that would make sure the probe got to Mercury slowly enough that Mercury would catch it, not the Sun. Cool, huh?

Anyway, Uchitel reintroduced us to the Pythagorean theorem, and then told us something I had either never learned or never listened to: in geometry, this theorem only applies to right triangles. Don’t try it on an isosceles or scalene triangle, missy! Or — horrors! — an equilateral triangle. It just won’t work!

That, however, didn’t stop Uchitel’s pre-calc students from trying it. Note to self: do not repeat this foolish mistake next semester.

I’m not surprised, though, that triangles are so important. Triangles are rather mystical in our culture. From the pyramids to the holy trinity to plain old-fashioned love triangles, I think we all, deep down, appreciate how important triangles can be.

I plan to master their various concepts for this course, and hope the knowledge will serve me well in the future.

*All text copyrighted by A.K. Whitney, and cannot be used without permission.
*

I’m glad your professor is focusing on the importance of triangles! The solution to so many geometry problems (and many others) is to break things down into triangles. If you’ll allow me to digress…

Imagine that you have 3 very strong, rigid beams, all connected by very flexible joints. You’ll find that you can’t change the shape of the triangle without breaking either the beams or the joints!

Then imagine you have 4 very strong, rigid beams, connected into a quadrilateral by very flexible joints. You’ll find that you can squish that quadrilateral flat as a bug. To prevent this from happening, you could add in a diagonal, which splits the quadrilateral into triangles. All of a sudden, no more squishing! This also works more generally, for shapes with much more sides.

So in some sense, triangles are a naturally “strong” shape. One of the many neat math applications I learned in my engineering days. 🙂

That is very cool! Thanks for sharing that. And I will remember your breaking things down into to triangles tip for solving proofs. Thanks again!