Revisiting yesterday’s challenging geometry problem

Yesterday Problem #18 from the 2008 AMC 10a gave my older son a bit of trouble. Unfortunately we were a little pressed for time so we didn’t get a chance to go through the problem in detail until today.

The problem is easy to state:

A right triangle has perimeter 32 and area 20. What is the length of its hypotenuse?

My son’s original approach was algebraic. It turned out that bringing the ideas from algebra and geometry together in this problem was difficult for him, so I started off today’s talk about the problem by reviewing the algebraic approach.

 

With the algebraic approach hopefully a little more clear than it was yesterday I moved on to a geometric approach to this problem. That approach involves a connection between area and perimeter of triangles that you learn from studying some properties of a triangle’s inscribed circle.

 

After talking about the general ideas you learn about area and perimeter from studying the triangle’s inscribed circle, we applied them to the specific triangle in the problem. It turns out that for right triangles the inscribed circle has a few extra properties that are really helpful in solving this problem:

 

Finally, since the last two pieces of this project had lots of different ideas, we went back to the beginning to make sure that they at least made a little sense:

 

I really like this problem. The algebraic approach to the solution is instructive. Although the algebra is a bit difficult, the exercise of bringing algebra and geometry together seems useful. The geometric approach provides a nice opportunity to introduce / review some beautiful ideas from geometry. Happy to have had the opportunity to go through this problem today in a non-rushed way.

Dave Radcliffe’s neat “linearity of expectation” tweet

Learned something from a Dave Radcliffe last week:

Linearity of expectation is a wonderful thing. https://t.co/D7nVCrOeOi

— David Radcliffe (@daveinstpaul) December 11, 2015

The link in his tweet goes here:

https://twitter.com/notch/status/675273039991345154%5Bembed%5D

I had to dig a little deeper to see what Dave meant about linearity of expectation, but when I did I found an incredible solution to the problem in just a couple of tweets:

Prior to this series of tweets from Dave, probably the most interesting “linearity of expectation” example that I’d seen was a new-to-me proof of the Buffon Needle problem in Jordan Ellenberg’s How not to be Wrong.

Although I don’t have the exact reference in Ellenberg’s book (I have the audio book version),  Lior Pachter has more or less the same neat explanation on his blog. It is amazing to me that changing the needle to a circle solves the problem in a snap!

Buffon’s Needle Problem on Lior Pachter’s blog

Anyway, a couple of points about problem in Dave’s tweets.  First, it wouldn’t have occurred to me to use linearity of expectation to attack this problem, and that is definitely my bad.  Dave’s example really showed me the power of the approach.  Second, my intuition for an approximate answer to the question was off by miles!  It actually took a while for me to understand how the number could be as high as 488  (I mean, you’ve got a 50% chance to win after about 500,000 tries, and, really, how many different people could have won 10 times by then . . . .), but I’m glad that Dave’s tweet made me think about this problem – I definitely needed the intuition adjustment!

Pretty incredible what you can learn on Twitter 🙂