“When am I EVER going to use this?” I’m sure that’s a question you’ve asked yourself numerous times as you pore over old math concepts that you felt sure you’d left behind after you passed the test you needed to in high school. I’m sure once you’re done with this test you’ll once again put this knowledge you learned into some kind of deep-freeze long-term storage that you hope you’ll never have to retrieve. Maybe that’s not the best course of action, because you never know when you’ll need it.
I was watching the finale of MTV’s “The Challenge: Rivals 2” a couple weeks ago. It’s a show much more well-known for contestants (most of whom are veterans of MTV’s “The Real World”) drinking, fighting and competing in painful physical challenges for the chance to win prize money than for math. Sandwiched somewhere between paddling canoes, eating piles of disgusting foods and hauling weighted bags was this problem:
A right triangle has side lengths of 75 and 168. What is the length of the hypotenuse?
The penalty for choosing the wrong answer was that the contestants would have to cut five ropes instead of one, and in a timed race where speed was critical, that was a mistake the teams could not afford to make. Commenters across the internet sympathized with the teams, saying they never would have been able to solve such a difficult problem without a calculator or pen and paper. Let’s put that to rest and see why this problem shouldn’t have been so tough.
First, since no one was given pencil and paper, the assumption shouldn’t have been that the problem was impossible. The assumption should have been that there was a shortcut. Second, whenever you see ugly numbers in a right triangle, look for Pythagorean triplets!
In this case, the numbers don’t fit perfectly into a 3:4:5 ratio or a 5:12: 13 ratio, but that doesn’t mean they’re useless. In fact, let’s look at that 5:12:13 ratio. Using that side length of 75, if this were a 5:12:13 triangle, the side lengths would be 75:180:195, which is our ratio multiplied by 15: 5*15=75, 12*15=180, 13*15=195.
Let’s use that. Since rather than 75 and 180, we have side lengths of 75 and 168, the one thing we know for sure is that our hypotenuse will be shorter than it would have been in the 75:180:195 triangle, so we can eliminate all answer choices greater than 195. That gets rid of three choices! All we have left is 154 and 184.
Use one more bit of logic. We know that the hypotenuse is always the longest side in a right triangle, which means that 154 has to go and we’re left with only one choice, the right one: 184.
Using Pythagorean triplets and a little bit of logic, this seemingly impossible problem suddenly looks very easy. Don’t panic when you face a tough problem. Take the time to think through it and you may find an excellent shortcut just like this one!