I hate trick questions.

In MATH 106 we have been studying uses of the Pythagorean Theorem in order to measure things with arcs, such as suspension bridges or bowed railroad tracks. Basic geometry, not too difficult:



So, in Thursday night's homework, there was this word problem:

"A 150-foot rope is suspended at its two ends from the tops of two 100-foot flagpoles. The lowest point of the rope is 25 feet from the ground. What is the distance between the two flagpoles?"

It seemed pretty straightforward, not much different than any of the other homework problems I had been working on. So I drew myself a little diagram, so that I could visualize the solution:

Based upon the values given, I could determine that:

Now I knew that c was actually a little less than 75ft, because of the sag in the rope. But we were instructed by our textbook to do these kinds of problems this way, knowing that there would be a little bit of error built in. Normally for these kinds of problems, it works pretty well. (Though not this time).

With the values of b and c known, I can now solve for a, which I know to be 50% of the length of the distance between the two flagpoles. In other words, the distance between the two flagpoles is 2 times the value of a.

So I plug my values into the Pythagorean Theorem. Since I am solving for a, I rewrite the formula, and solve:



Uh oh... a² is the square root of zero? That can't be right. Something is very, very wrong here!

I go back and re-do all my calculations. I double check everything. I re-read the problem. I re-evaluate my initial assumptions. I even re-draw the diagram!

And every time I get the same result!

I ended up spending almost 2 hours on this one problem Thursday night, and didn't end up getting to bed until almost 2 am because of it. I had completed all of the rest of the homework assignment, except for this one problem. So I finally decided just to sleep on it.

Friday morning before class, I went into the Learning Resource Center on campus and asked for a math tutor's help. A nice guy who had helped me before sat down with me, and listened as I explained the problem to him. We went over the instructions in the textbook on how to solve this kind of problem. He went over all of my calculations... and came up with the same results as I did!

He went and got another tutor, who also went over all my work... and came up with the same thing. What was going on here?

The second tutor said, "Despite the fact that the book is telling you to solve the problem this way, I don't think this problem is solvable with the Pythagorean Theorem."

So, now having spent more than 4 hours of time on this one stupid homework word problem, I went to class, prepared to confront the TA who teaches the class.

Turns out I wasn't the only one who was stumped by this problem.

I asked the instructor to show us how to use the Pythagorean Theorem to solve this word problem. Her response:

You don't need use it to solve this problem.

The only way you can have a 150ft rope, suspended between two flagpoles, where the rope hangs down to 25 ft above the ground is...

(Are you ready for this?)

...If the flagpoles are standing right next to each other, with no distance between them.

D'OH!

You know, for someone who prides himself on being able to think outside the box, I didn't see that one coming.

I mean, in hindsight, the math told me that the distance between the two flagpoles was zero (2 times a, with a being the square root of zero)!

But neither I, nor the two tutors helping me, nor half my classmates, were able to make the intuitive leap from that unexpected result to the actual truth.

I hate trick questions.

Especially when I get them wrong.