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!
Imagine you are given a question that describes the following situation:
There is an isosceles triangle inscribed in a circle with radius = 5. The center of the circle is at (0,0). The triangle is symmetrical about the y-axis.
Do you have a picture in your head about what that would look like? If the answer is no, it could be because you’re unfamiliar with some of the terms that were described. Perhaps you’ve forgotten what an isosceles triangle is (a triangle with two sides of equal length) or perhaps you don’t quite remember which one is the y-axis (it’s the vertical one). If that’s the case, it’s time to go back and review some of the core mathematical terms that you’ll need to know in order to be successful on your test. The Barron’s video course for your test would be a great place to start.
However, even if you got that far, there’s potentially another problem. My guess is that most of you have imagined something like this:
But did you also consider that this green triangle is a possibility as well?
The common advice we give when figures in the coordinate plane is that you should draw a picture. This is sound insofar as your brain can much more easily interpret the graphical information than a set a words given to describe that information. However, draw a picture can get you into trouble when you need to draw the picture or even the pictures.
It’s a word of caution that I hope you’ll remember. When you’re given some graphical information in word form it’s great to translate that into a picture. However, make sure you really take the time to dissect what all of the information means and could mean, so that you aren’t overlooking a part of the solution.
Good luck and happy studying!
Today we’re going to take a look at a shape that appears extremely often on the geometry portion of your test: the isosceles triangle. Since geometry boils down to circles and triangles (and the lines and curves that define them), understanding this very common triangle type is essential. Take a look at this image which shows the essentials of an isosceles triangle:
An isosceles triangle is one where at least two of the sides are equal in length. The specific case where all three sides are equal in length is an equilateral triangle and the implications of that are more widely known, so we’ll reserve our current discussion to the case where two sides are equal in length and the third is different.
The reason that isosceles triangles are so widely used is that they allow you to infer a large amount of information from a relatively small number of given facts. For instance, in the figure above if we’re told that angle A measures 36 degrees, we can solve for the measures of both angles B and C. Since the length of sides AC and AB are equal, the measures of angles B and C must also be equal. So, solving for the measure is simply this:
36 + 2(AB) = 180
Another great thing about isosceles triangles is that it works both ways. If rather than knowing anything about the sides, we knew that angles B and C were equal in measure we would still be able to infer that sides AC and AB were equal. That gives lots of flexibility to the test maker. He can present you with the sides or the angles and rely on you to get the rest of the information.
The last great thing to know about isosceles triangles is that they have a line of symmetry that bisects the non-equal angle and creates two right triangles. See the figure below:
So, not only do isosceles triangles have their own unique and helpful properties, but they can easily be turned into right triangles which are extremely helpful as well. Keep these ideas in mind as you work through the geometry section of your next test!
Multiple figures geometry questions are common on the GMAT. Make sure you recognize how the two shapes relate to each other by noting
When you see a 60 degree angle in a triangle, you should be thinking one of two possibilities: equilateral or 30-60-90 triangle. Once you spot the latter in this problem, the rest of solving just comes down to arithmetic. Be on the lookout for problem types that show up often on the GMAT!