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GMAT Geometry (Data Sufficiency)
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Author:  questioner [ Fri Aug 20, 2010 4:09 am ]
Post subject:  GMAT Geometry (Data Sufficiency)

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In the figure above, three segments are drawn to connect opposite vertices of a hexagon, forming six triangles. All three of these segments intersect at point A. What is the area of the hexagon?
(1) One of the triangles has an area of 12.
(2) All sides of the hexagon are of equal length.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(E) Statement (1) is not sufficient because we do not know whether the hexagon is a regular hexagon (meaning that all 6 sides
have the same length). We could only calculate the area if the hexagon could be divided into six congruent triangles.

Statement (2) is also insufficient because we are not given any numbers. So, it is impossible to calculate the area. Combined, the two statements are still insufficient. A hexagon could have six equal sides, yet not be a regular hexagon. Imagine pushing opposite sides of a regular hexagon closer to each other: the
hexagon would flatten, while all the sides would remain the same length. One of the triangles within this flattened hexagon could have an area of 12, but not all of the triangles would have the same area. Therefore, we can't calculate the area, even by combining the statements, and the correct answer is choice (E).
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In my opinion the the answer should be (A).
I will attempt to give a detailed explanation below:
Since it is a Hexagon, sum of internal angles is 720 using the formula: (n – 2) × 180, where 'n' is the number of sides of the polygon.

Therefore each internal angle is 120 degree. From this I say each triangle formed inside is an equilateral triangle. From (1), the area is 12 for each triangle. This is SUFFICIENT to conclude the hexagon having six triangles having an area of 12 each, has an area of
72.

Please confirm
.

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Author:  Gennadiy [ Fri Aug 20, 2010 4:48 am ]
Post subject:  Re: math (test 1, question 25): geometry, data sufficiency

When you solve any data sufficiency geometry question you should avoid graphics distorting your reasoning.

In this case the hexagon seems to be regular (all sides and angles are equal) and it affects the reasoning. Therefore you may conclude that all angles are equal (120⁰) but we have no facts that this conclusion can be based on.

The question statement gives us just the fact that we have a hexagon. It is just some arbitrary hexagon and can look in many different ways:
Image
or
Image
and many other different shapes.
That gives us a clear idea why statement (1) is insufficient.

Statement (2) also doesn't imply that hexagon is regular (regular hexagon must have all equal sides AND angles). If we know that all sides are equal then hexagon can still have diverse shapes:
Image
or
Image
and many other variants.

So be careful with geometry graphics and always check if what you see is based on facts.

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hexagon4.gif [3.52 KiB]
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Author:  questioner [ Wed Apr 17, 2013 7:59 am ]
Post subject:  Re: GMAT Geometry (Data Sufficiency)

Hi,

As per my understanding, every polygon with equal sides is always cyclic in nature.

Could you, please, correct if I am wrong.

Best regards.

Author:  Gennadiy [ Wed Apr 17, 2013 7:59 am ]
Post subject:  Re: GMAT Geometry (Data Sufficiency)

Quote:
As per my understanding, every polygon with equal sides is always cyclic in nature.
A polygon with equal sides is cyclic if all its angles are equal. In other words such polygon is a regular polygon (all sides are equal and all angles are equal).

Here is an example of a polygon with equal sides but not with equal angles. It is not cyclic.
Image

Author:  questioner [ Wed Apr 17, 2013 8:00 am ]
Post subject:  Re: GMAT Geometry (Data Sufficiency)

On the gmat, will it specifically state whether the hexagon is a regular
hexagon or not? I assumed that when you mentioned hexagon it was a regular
hexagon.

Author:  Gennadiy [ Wed Apr 17, 2013 8:01 am ]
Post subject:  Re: GMAT Geometry (Data Sufficiency)

With the problem at hand you don't need to know if the hexagon is regular or not. All of the necessary information is provided. However, on the GMAT when you actually encounter a problem that involves a geometric figure that is regular, and, it is required that you know the latter in order to solve the problem, then yes, the GMAT question will provide you with that information.

Take care.

Author:  questioner [ Wed Apr 17, 2013 8:01 am ]
Post subject:  Re: GMAT Geometry (Data Sufficiency)

I understand the question, but I think "three segments connecting opposite vertices of a hexagon intersect at a point" means that the hexagon is a regular hexagon!

Author:  Gennadiy [ Wed Apr 17, 2013 8:02 am ]
Post subject:  Re: GMAT Geometry (Data Sufficiency)

questioner wrote:
I understand the question, but I think "three segments connecting opposite vertices of a hexagon intersect at a point" means that the hexagon is a regular hexagon!
No. You can see that if you draw various cases of 3 arbitrary segments that intersect in one point, then you just need to connect the ends of the segments to create various hexagons.

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Author:  questioner [ Thu Jun 20, 2013 4:03 pm ]
Post subject:  Re: GMAT Geometry (Data Sufficiency)

If all sides are equal, its a regular hexagon. An area of a regular hexagon can be considered 6 × area of the equilateral triangles. Thus using both the statements we could arrive at the solution . Hence option C is correct.

Author:  Gennadiy [ Thu Jun 20, 2013 4:08 pm ]
Post subject:  Re: GMAT Geometry (Data Sufficiency)

questioner wrote:
If all sides are equal, its a regular hexagon.
A hexagon is regular if all sides are equal AND all interior angles are equal.

Think of 6 equal sticks connected in form of a hexagon and imagine moving those sticks. Here is an examples of a non-regular hexagon, which sides are all equal:
Image

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