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21 Jul 2017, 05:01
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D
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Difficulty:
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44% (02:08) correct
56%
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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.
M36-139
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17 Nov 2021, 09:12
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Bunuel
Official Solution:
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.
Consider regular polygons below:
Notice that when we increase the number of sides, the ratio of the longest diagonal to the side increases. So, more sides the larger the ratio.
Let's check what this ratio is for a 6-sided regular polygon, hexagon:
The degree measure of \(\angle AOB = \frac{360°}{6}=60°\). Since AO = BO, then \(\angle ABO = \angle BAO = \frac{180° - 60°}{2}=60°\). So, triangle ABO is equilateral. Thus, AB = AO, but AO is half the diagonal, so \(diagonal:side=2:1\)
Since we are given that the ratio for our polygon is more than 2 and we know that the ratio increases as we increase the number of sides, then our polygon must have more than 6 sides. Sufficient.
(2) The degree measure of an interior angle of polygon X is NOT an integer.
The degree measure of an interior angle in a regular polygon is \(\frac{(n-2)*180°}{n}\), where \(n\) is the number of sides.
So, we are told that \(\frac{(n-2)*180°}{n}\neq integer\). Notice that since 180 is divisible by 3, 4, 5, and 6, then for these values of \(n\), \(\frac{(n-2)*180°}{n}\) will be an integer. Thus, the number of sides, \(n\), must be more than 6. Sufficient.
Answer: D
General Discussion
21 Jul 2017, 05:52
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IMO D.
Statement 1: As we move from Triangle to hexagon we can see that this ratio increases from 1 ----> 1.732, hence for ratio to be 2, the polygon should be the one with more than 6 sides
Statement 2: As we move from triangle to hexagon we can see that the interior angle will always be integer. But as soon as we have more than 6 sides the interior angle becomes non Integer.
Mathematically, any interior angle can be given by [ ( N-2)/N x 180], where N= number of sides of the regular polygon. All values of N from 3 to 6 are factor of 180.
Statement 1: As we move from Triangle to hexagon we can see that this ratio increases from 1 ----> 1.732, hence for ratio to be 2, the polygon should be the one with more than 6 sides
Statement 2: As we move from triangle to hexagon we can see that the interior angle will always be integer. But as soon as we have more than 6 sides the interior angle becomes non Integer.
Mathematically, any interior angle can be given by [ ( N-2)/N x 180], where N= number of sides of the regular polygon. All values of N from 3 to 6 are factor of 180.
