Imagine This

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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!

 Helpful Isosceles Triangles

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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!

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 A Right Triangle Peg In A Round Hole

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Multiple figures geometry questions are common on the GMAT. Make sure you recognize how the two shapes relate to each other by noting

Question of the Day

Evaluate the expression if  and are positive numbers in a geometric progression and and .

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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!

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 Avoiding Distracting Information

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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?

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Much of the information presented at the outset of this problem is unnecessary, and is presented simply as a means to confuse and overwhelm. Focus on the pertinent data in every problem, and don’t let an excess of information throw you off track!