# So a Circle Walks into a Bar…

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!

# Which Lane to Choose– Part III

In our last post, we explored a curious problem about which lane to choose in traffic and found some surprising correlations to math. However, when we last saw our equation we were stuck with too many variables, so now we return to address that problem of how to find the measure of the two central angles.

In the real world, the answer to this question might be to get out some sophisticated equipment to measure this angle. That won’t be the approach that we’ll take because on a standardized test there isn’t a way to acquire new information. You have only what you’re given on the page. That can be an advantage, however, because limiting the number of potential inputs means that the same types of tricks and techniques you use on one problem are likely to be applicable to things you see in the future. In this case, the trick we’ll use is making the angles the same.

As I said before the measure of the central angle leading to the outside ring will be smaller because the cars are not backed up quite as far in that lane. However, in order to take advantage of the information we have, let’s use the same angle for both arcs, and compensate for the open space in the outer arc later. If we do that, we get:

$Difference=2\pi r_{1}*\frac{\theta}{360}-2\pi\left (r_{1}+12\right )*\frac{\theta}{360}$

Since we now have our terms lining up in a way that will allow them to cancel out, we just need to deal with that shorter outer arc. The measure of the outer arc will be shorter than the equation listed above by the effective length of one car, that we estimated above at 20 feet. So, feeding that into our equation we get:

$Difference=2\pi r_{1}*\frac{\theta}{360}-\left[2\pi\left (r_{1}+12\right)*\frac{\theta}{360}-20\right]$ $Difference=2\pi r_{1}*\frac{\theta}{360}-2\pi r_{1}*\frac{\theta}{360}-2\pi\left(12 \right)*\frac{\theta}{360}+20$ $Difference=20-24\pi*\frac{\theta}{360}$

Since we set this up as the inside lane minus the outside lane, any positive difference would mean that the inside lane had to travel a longer distance, and thus the outside lane would be the better choice, or vice versa. Calculating the break-even point (where either lane would have to travel the same distance) is:

$24\pi*\frac{\theta}{360}=20$

Solving gets us:

$\theta\approx 96^{\circ}$

So, if the measure of the central angle that cuts the arc is greater than 96 degrees, the inner lane is the wise choice. However, if the measure of the central angle that cuts the arc is less than 96 degrees, the outer lane is best. Having traveled this road many times, I know that this long on-ramp cuts an arc longer than 96 degrees, and thus I (luckily, since I couldn’t do all these calculations on the fly) made the correct choice in taking the inner lane.

Next time, we’ll wrap this up by asking what all of this means.

# Which Lane to Choose?– Part I

I was stuck in traffic the other day, and found myself with a choice. I was getting on the freeway where the on-ramp is a long, smooth curve with two lanes. The inner lane was backed up about one car length more than the outer lane. Being on the verge of being late (as I so often am), and not being able to see much more than the few cars ahead of me, I had to make the choice given the information that I had: which line was shorter?

As far as consequential decisions go, this one ranks pretty far down the list. However, being stuck in traffic is an excellent time to hone the critical thinking skills that will allow you to solve the less straight-forward quantitative problems that you might face on the GMAT or GRE or even occasionally the SAT.

The proper approach to solving any problem is first to identify the goal. In this case, the goal is to determine whether the distance my car would have to travel in the inner lane would be greater than or less than the distance my car would have to travel in the outer lane.

Next, I need to assess what information I have that I can use to solve this problem. I know that the average width of a lane is 12 feet. I know that an average car is about 16 feet in length, and if I include a reasonable gap between myself and the next car in stop and go conditions, each car on the road is going to occupy about 20 feet. I also know that any smooth curve can be thought of as a piece of some larger circle.

That let’s me refine my task and start to work with this information a little more fully. Essentially, I’m trying to measure the differences between the lengths of arcs on concentric circles. But, does that really help? I don’t know exactly how large the circle would be, or exactly how big a slice of that circle I’m cutting. Rather than despair, the key now is to write down the formula and see what information you have, and what information you need.

$\textrm{Arc Length}= 2\pi r\times \frac{\theta }{360}$

This equation tells us that we have two variables, the radius and theta (the measure of the central angle that cuts the arc). We don’t have means to acquire these numbers short of renting a helicopter and getting some perspective on the situation. And let’s be honest, if I had access to a helicopter I wouldn’t have been sitting in traffic in the first place. So, is it back to despairing?

# Avoiding Distracting Information

In addition to testing your knowledge of difficult topics and challenging concepts, the GRE may also try to overwhelm you with an excess of information or complexity.

In the following question, what information is useful, and what is presented simply to distract you from the problem at hand?

## Question of the Day

A geometrical figure   is defined such that there are concentric semi-circles with a right isosceles triangle inscribed within each semi-circle. The following figure shows with 3 concentric circles and 3 right triangles, one inside each semi-circle. If the radius of the th innermost circle is , pertaining to the figure, where radius  is given to be  units, what will be the perimeter of the 4th triangle?