Wed 26 Jul 2006

## Triangles

Posted by **David Dudley Field '25** under **Alumni** at 7:23 am

Diana Davis ’07 seems to be having a fun time teaching math this summer. She reports:

We had a problem in geometry a few days ago that involved saying whether something was an acute triangle or an obtuse triangle. This is an easy problem, because if one of the angles is greater than 90 degrees, you say “obtuse,” and if none of the angles is greater than 90 degrees, you say “acute,” and if two of the angles are greater than 90 degrees, you say “get back to plane geometry.”

This problem got me to thinking: what are there more of, acute triangles or obtuse triangles? Which naturally led me to wonder: what is the ratio of acute triangles to obtuse triangles?

Now, one way you could do the problem would be to make a “simplification” like say that the angles have to be whole numbers, and then count. But then you’d have to do a lot of counting, and you’d get all confused with the right triangles. So this would be a bad way to solve the problem.

The way I decided to solve the problem was to imagine that the first angle is plotted on the x-axis, the second angle is plotted on the y-axis, and the third angle is plotted on the z-axis. Then your x, y, and z values can all range between 0 and 180, but with the constraint that x+y+z=180. This is the equation of a plane that intersects the axes at (180,0,0), (0,180,0), and (0,0,180), respectively.

Interesting stuff, but a diagram would be very nice. Perhaps Brent Yorgey ’04 can chime in. Diana also

realized recently how much free time we have here — unless we have duty, we have all of Wednesday afternoon free, and Saturday afternoon and almost all of Sunday. So I could go somewhere even if it wasn’t my weekend off. For example, I could up and decide to go to Williamstown, you know, tomorrow.

Or you could post some pictures of Williams. Having tricked all of your EphBlog fans into addiction, you owe us a couple of summer-time fixes. Don’t make us beg!

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## 2 Responses to “Triangles”

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Brent says:

Ask and ye shall receive. =) See my comment on Diana’s post. Making 3D diagrams is difficult, BTW, without using some sort of software like Mathematica.

July 26th, 2006 at 11:47 amSam Jackson says:

Hello again EphBlog! I posted here, oh, many months ago, prattling about something insignificant. I was told to drop back again when Williams came up during my college app. process, so here I am… :) I got my Williams marketing material–err, viewbook–the other day, very posh, very glossy. I’d love to share the pictures from my tour at the beatiful campus, but… I haven’t toured. I don’t know if I will (or if I will apply… hmm) but western MA is just a little out of the ways from Boston to hit up sans drivers license for just a casual jaunt. Anyways I will thoroughly present my thoughts on what I’ve got (um, viewbook?) on the website I made for my college process-tracking, samjackson.org/college. Shameless plug notwithstanding, I will drop back next time Williams visits my school (Exeter) and report on what I’m sure is the engaging friendliness of your Admissions staff!

July 26th, 2006 at 5:19 pm