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 a 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

My question: 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.

With the problem at hand, AEK, 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.

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!

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.

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.

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:

or

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:

or

and many other variants.

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

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.

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.

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:

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