So a Circle Walks into a Bar…

5

It sounds like the beginning of a math joke, but it isn’t.

“So a right triangle is inscribed into a circle…”

That’s the premise of a couple interesting GMAT questions that I came across lately, so I thought I’d share the issues that these problems bring. First it’s important to define that term inscribed. It’s the kind of term that you may have come across several times without ever knowing what it means because the visual diagram that accompanies the problem has you covered. In geometry, when we talk about something being inscribed we mean that it is drawn inside another shape such that all of its corners touch the edge of the larger shape without going outside of it. When a shape is inscribed within a circle it’s a little like that shape has a custom-built bubble surrounding it.

Now back to our problem. So there’s a right triangle in a bubble. So What? Well that particular situation actually gives us a very important piece of information. Whenever a right triangle is inscribed in a circle, the hypotenuse of the triangle is the diameter of the circle. That’s a fantastic rule, and one you ought to remember, but when we get to the difficult end of the quant section where a question like this is likely to occur, we’re probably going to need more than that.

So what other concepts fit in with this rule? Well, our rule gives us a fantastic way to find the hypotenuse of the triangle if we know something about the circle (or vice-versa), so a nice extra step is when the GMAT asks about the length of one of the other sides of the triangle. When would we be able to find the length of the other sides of the right triangle knowing only the length of the hypotenuse? When it’s a special right triangle! So, be on the lookout for 30:60:90 triangles or 45:45:90 triangles. Even if these aren’t immediately apparent, remember that every distance from the center of the circle to the edge of the circle is a radius, and drawing one or more of these radii in often gives you more opportunity to solve.

Keep this fantastic rule and these tips in mind the next time you come across a similar problem!

 Reality TV and Right Triangles

44

“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?

 

238

154

206

184

196

 

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!