Which of the following lists the number of points at which a circle can intersect a triangle?

(A) 2 and 6 only

(B) 2, 4 and 6 only

(C) 1, 2, 3 and 6 only

(D) 1, 2, 3, 4 and 6 only

(E) 1, 2, 3, 4, 5 and 6 only

I'm often asked: "What's the most important thing to work on in order to succeed on the GMAT". Not a simple question. Of course you need all of the quant and verbal fundamentals. And you're going to need to learn GMAT specific strategy. Above all that though is what I think is the most important quality: organized curiosity. You have to be willing to explore a bit. Not randomly though. In an organized way. Conducting mini experiments to feel out a question. This geometry question from the GMAT prep tests is a good example of the type of GMAT puzzle that rewards those willing to try a few things. Most often students stare at this one and refuse to play a little bit drawing out scenarios. Time pressure gets the best of them and then it's over. You have to ignore the clock a bit so you can let some ideas flow. GMAT time pressure is real. In that it's a timed exam and for most people that adds a formidable constraint. Still, if you're going to spend precious time working on a question better make that quality time leaving yourself open to the type of thinking that tends to lead to success. Developing the habit of organized curiosity requires taking the risk of ignoring the clock. Hopefully you'll find that by opening things up a bit you think more flexibility and creatively and give yourself the time to organize things properly so that you actually are not only more effective but more efficient. This isn't a skill that you just turn on. It needs to be practiced. And while you're practicing you're probably going to fail a whole bunch. But remember to keep practicing how you actually want to perform. It's true that in the short term you might be able to get away with lower quality thinking and make some gains refining that but if you're looking to fundamentally change your GMAT performance you likely need to not only work on GMAT content by your general approach to problem solving. Diagram below with the solution and here's a video explanation: Which of the following lists the number of points at which a circle can intersect a triangle # The perimeter of a certain isosceles right triangle is 16 + 16√2. What is the length of the hypotenuse of the triangle? Three ways to solve this beast and a video solution!

(A) 8

(B) 16

(C) 4√2

(D) 8√2

(E) 16√2

A lot of people get this GMAT geometry question from the GMAT prep software wrong. It's true that the algebra gets a little tricky but conceptually there's not much going on. We just need some basic triangle rules and vocabulary (perimeter, isosceles, right triangle...) And, to add, there are three ways to solve this thing. You can tackle the algebra, plug in answer choices, or you can even estimate. We're going to take a look at each solution below.

You need to know that the sides of an isosceles right triangle are always in the ratio x, x, x√2. No excuses for not knowing GMAT basics. You have to be fluent with the fundamentals so that you can focus on what the test is really about: critical thinking.

The perimeter of this isosceles right triangle then is 2x + x√2. And. according to the given information, that perimeter equals 16 + 16√2. About half of people stop here and stay: Bingo! The hypotenuse of the triangle must be 16√2!!! Nope.

That's some fuzzy logic. Let's keep simplifying by trying to isolate the variable. You can do that by factoring out the x. Do you know how to get rid of radicals in the denominator (rationalizing)? Well, that's another basic that you'll need to conquer the GMAT quant. Keep in mind that we're solving for x√2 (the hypotenuse). See the diagram below for the worked out solution. ## You can also use the answer choices

I'd still do a bit of simplifying but then you can plug in the answers and see if the equation stays balanced (what is on the left equals what is on the right). Again, keep in mind that the answers are the hypotenuse, x√2, not x. So in order to plug them in for x we need to divide them by √2. Sounds more complicated than it is. Plugging in answers ends up being even easier than the algebra. ## Last but not least you can estimate

Estimation is a very helpful tool for getting through some GMAT questions. My advice is:

1. Keep your approximation/estimation in line with how far apart the answer choices are. In this case our approximations are within 10% and the closest answers are about 30% apart giving us some breathing room.
2. Use estimation/approximation as a useful tool deployed when appropriate not as a crutch for when you're in panic mode.

Here we just have to call √2 1.4. After that the arithmetic is basic. You can multiply everything by 10 to get rid of the decimals and then simplify. 16/17 is so close to 1 that we can just call it 1. 160/17 is very close to 10. With that we get 17. But we know that we're a tiny bit less than 17. The only answer that's remotely close is 16. 16√2 is almost 24. 8√2 is little less than 12. Those are way too far off to be considered. ## More Challenging GMAT Geometry Examples Involving Isosceles right triangles

This is one of the toughest GMAT questions ever. In a lot of ways the solution is very basic but most people don't get there or even close. This one is a notch tougher than the one above and I really don't expect most people to get it correct. I'd be 50/50 on this if I saw it on an exam.

Still, it's a great learning question as it can reveal that the quant part of the GMAT is simple. There's no fancy math here. It's all about understanding the given information and the question and getting it all organized. In the figure above, each side of square ABCD has length 1, the length of line segment CE is 1, and the length of line segment BE is equal to the length of line segment DE. What is the area of the triangular region BCE?

This one doesn't involve right isosceles triangles but I think it's very similar because the geometry aspect is basic but the follow through and algebra gets a little tricky: A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?

Here's another right isosceles triangle question from the GMAT Official Guide that's very similar in difficulty to the above. In the figure above, is the area of triangular region ABC equal to the area of triangular region DBA?

And finally a similar right isosceles triangle question from GMAT question of the day.

Happy studies!

# GMAT Question of the Day - Problem Solving - Geometry

If s is between 0 and 9, how many different equilateral triangles with side s can be formed that have an area which is an integer value?

A. 3

B. 4

C. 5

D. 6

E. 7

## GMAT Question of the Day Solution

This GMAT question of the day is tough but not impossible. What makes it tough? You have to consider a few different things while keeping in mind the limitations imposed by the question. First off - what's the area of an equilateral triangle? Now consider what numbers will create an integer value for the area? We have to get rid of the 4 in the denominator and the √3 in the numerator. Let's look at one issue at a time:

1. To cancel the 4 you need to have a at least a 4 in the numerator. It doesn't need to be a 4 but could be an 8 or a 12 or a 16. Anything with at least one 4 as a factor. Keep in mind the limitation that S must be less than 9. So that leaves 2, 4, 6, and 8 each of which when squared will have at least one 4.

2. We also need to either cancel or transform the √3. We can do this by either dividing or multiplying by √3. This means that S must have 3^1/4 in the numerator or the denominator.

Putting both together you have these options for S:

2(3^1/4), 4(3^1/4), 6(3^1/4)

2/(3^1/4), 4/(3^1/4), 6/(3^1/4), 8/(3^1/4)

Note that 8(3^1/4) will not work because it is greater than 9. That leaves seven different triangles. # GMAT Question of the Day - Problem Solving - Geometry If DF I| AC and x = 1.5y, what is the ratio of the areas of triangle ABC and triangle DEF?

A. 4/7

B. 3/2

C. 7/4

D. 9/4

E. 16/3

## GMAT Question of the Day Solution

The rules governing similar triangles are relatively basic. So what's challenging about GMAT similar triangles questions?

1. Spotting that the question is in fact testing similar triangles

2. Organizing the information so that it is clear which angles and sides are corresponding.

How can you make GMAT similar triangles questions easier? First thing, whenever you see triangles that share sides or triangles within triangles consider that the triangles may be similar. Remember that you only need two angles to be the same in order to prove similarity. One thing that can help to keep things organized is to label the angles. You don't need to know that actual values. Just giving the angles variables will help illustrate which angles are in fact the same and which sides correspond with one another.

In this case the two triangles must be similar because two of the angles are the same. You can infer this information by the fact that DF and AC are parallel (parallel lines cut by a transversal). Now you can substitute x for y and find the ratio of the sides. Here's a GMAT shortcut: the square of the ratio of sides is equal to the ratio of the areas.

# GMAT Question of the Day - Data Sufficiency - Geometry

A hardware store sells paint in two different containers. Container A is a cube with an edge of k. Container B is a right cylinder with a height of 5k. Assuming that each container is filled to capacity which container will cost less per unit of volume?

(1) Container A costs 1/3 as much as Container B

(2) Container B has an inside diameter which is less than 2k

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