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# Does regular polygon X have more than 6 sides?

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Re: Does regular polygon X have more than 6 sides? [#permalink]
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Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

Does regular polygon X have more than 6 sides?

(1) The ratio of the length of the longest line that can be drawn between two vertices of polygon X to the length of any one side of polygon X is greater than 2.
(2) The degree measure of an interior angle of polygon X is NOT an integer.

The regular polygon has
Exterior angle = 360/n = 360/6 = 60 degrees
Interior angle = 180-Exterior = 180-(360/n) = 180-60 = 120 degrees
Length of Diagonal = 2*Length of side

Statement 1: The ratio of the length of the longest line that can be drawn between two vertices of polygon X to the length of any one side of polygon X is greater than 2.
Since the longest diagonal in a hexagon is twice the length of side therefore the polygon discussed in Question is definitely not a six sided polygon
SUFFICIENT

Statement 2: The degree measure of an interior angle of polygon X is NOT an integer.
But we have already seen that the degree measure of a six sided polygon is 120 hence it's definitely not a six sided polygon
SUFFICIENT

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Re: Does regular polygon X have more than 6 sides? [#permalink]
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

Does regular polygon X have more than 6 sides?

(1) The ratio of the length of the longest line that can be drawn between two vertices of polygon X to the length of any one side of polygon X is greater than 2.
(2) The degree measure of an interior angle of polygon X is NOT an integer.

The question asks "Does regular polygon X have more than 6 sides?"

(1)
Let's take a square. The length of the longest line is the diameter of the square. So the ratio of diameter to the side increases because the length of the diameter is constant, and the length of the side (chord) decreases.

As the question is concerned with a 6-sided polygon, let's take a regular hexagon - 6 sided regular polygon.
The sum of its angles is (n-2)x180=720. So, each angle is 720/6=120 degrees.
The length of the longest line is the diameter of the circumscribed circle, i.e. AD.
Angle OAF = angle OFA= 60 degrees => triangle AOF is equilateral. So, AF is equal to the length of the radius => 1/2 of the diameter.
Per st.1, the ratio of diagonal is more than 2, so the polygon must have more than 6 angles.
Sufficient.

(2)
Angle of a triangle = 180/3 = 60
Angle of a square = 360/4 = 90
Pentagon: Sum of angles = (n-2) x 180 = 540. Angle of pentagon = 540/5 = 108
Angle of hexagon, as mentioned above, is 120 degrees.

Sufficient, we can stop here and move on.

The asnwer is D

Just to prove, the sum of angles of the 7 sided polygon (heptagon) = (7-3)x180=900. 900/7=128 4/7

Hence D
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Re: Does regular polygon X have more than 6 sides? [#permalink]
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

Does regular polygon X have more than 6 sides?

(1) The ratio of the length of the longest line that can be drawn between two vertices of polygon X to the length of any one side of polygon X is greater than 2.
(2) The degree measure of an interior angle of polygon X is NOT an integer.

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Re: Does regular polygon X have more than 6 sides? [#permalink]
GMATinsight wrote:
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

Does regular polygon X have more than 6 sides?

(1) The ratio of the length of the longest line that can be drawn between two vertices of polygon X to the length of any one side of polygon X is greater than 2.
(2) The degree measure of an interior angle of polygon X is NOT an integer.

The regular polygon has
Exterior angle = 360/n = 360/6 = 60 degrees
Interior angle = 180-Exterior = 180-(360/n) = 180-60 = 120 degrees
Length of Diagonal = 2*Length of side

Statement 1: The ratio of the length of the longest line that can be drawn between two vertices of polygon X to the length of any one side of polygon X is greater than 2.
Since the longest diagonal in a hexagon is twice the length of side therefore the polygon discussed in Question is definitely not a six sided polygon
SUFFICIENT

Statement 2: The degree measure of an interior angle of polygon X is NOT an integer.
But we have already seen that the degree measure of a six sided polygon is 120 hence it's definitely not a six sided polygon
SUFFICIENT

This post, I think does not fit here. The question asks for whether the polygon has more than 6 sides, while this reply by GMATInsight proves that it is not a Hexagon/6 sided polygon.
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Re: Does regular polygon X have more than 6 sides? [#permalink]
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

Does regular polygon X have more than 6 sides?

(1) The ratio of the length of the longest line that can be drawn between two vertices of polygon X to the length of any one side of polygon X is greater than 2.
(2) The degree measure of an interior angle of polygon X is NOT an integer.

For the 2nd option,

If the polygon has 12 sides,

interior angle = (n-2/n)*180 = (10/12)*180 = 150 which is an integer.

So only 1st option is correct according to me.
Let me know if I am wrong.
Thanks!
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Re: Does regular polygon X have more than 6 sides? [#permalink]
I have a question.
If it's a 10 sided poly then the interior angles are [(10-2)/10]=144 which is an integer so shouldn't A be the answer?
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Re: Does regular polygon X have more than 6 sides? [#permalink]
chinmaysawant wrote:
Bunuel wrote:

Fresh GMAT Club Tests' Challenge Question:

Does regular polygon X have more than 6 sides?

(1) The ratio of the length of the longest line that can be drawn between two vertices of polygon X to the length of any one side of polygon X is greater than 2.
(2) The degree measure of an interior angle of polygon X is NOT an integer.

For the 2nd option,

If the polygon has 12 sides,

interior angle = (n-2/n)*180 = (10/12)*180 = 150 which is an integer.

So only 1st option is correct according to me.
Let me know if I am wrong.
Thanks!

Take n= 7 or 11..

You won't get an integral value for the interior angles.. infact all the values of n between 3 and 6 will give an integral value, and n is nothing but the number of sides of the polygon... so n > 6 will give you atleast one non integral value of the interior angles.. and that's all what you need.

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Re: Does regular polygon X have more than 6 sides? [#permalink]
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Re: Does regular polygon X have more than 6 sides? [#permalink]
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