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!

quantQ
 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!

quantQ
 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!

quantQ
 Forget a formula? Just derive it!

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Despite your best efforts, you will probably not be able to remember every single fact about every single possible algebra or geometry property in the universe for the GRE. The good news – it doesn’t matter!

Beyond the simplest properties, most relevant information can be derived from what you know. When solving the following problem, see if you can do it without knowing the formula for a trapezoid.

Question of the Day


The figure shows two squares ABCD and PQRS having a common center O and having sides of 6 units and 4 units respectively. A semicircle passes through the center O. What is the area of the shaded region?

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Even if you don’t have the formula for the area of a trapezoid memorized, you can still solve this problem, because a trapezoid is made up of a rectangle and two triangles (the formulae of which you DO definitely need to know). So if you come across a fact or figure on a test that you can’t calculate, don’t freak out – see if you can use what you already know to solve the problem!