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# In the figure above, polygon N has been partially covered by a piece

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Math Expert
Joined: 02 Sep 2009
Posts: 53831
In the figure above, polygon N has been partially covered by a piece  [#permalink]

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08 Mar 2018, 21:24
00:00

Difficulty:

45% (medium)

Question Stats:

47% (00:59) correct 53% (01:26) wrong based on 74 sessions

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In the figure above, polygon N has been partially covered by a piece of paper. How many sides does N have?

(1) x + y = 45
(2) N is a regular polygon

Attachment:

Hidden_polygon_1.png [ 3.95 KiB | Viewed 763 times ]

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Posts: 53831
Re: In the figure above, polygon N has been partially covered by a piece  [#permalink]

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08 Mar 2018, 21:24
Bunuel wrote:
In the figure above, polygon N has been partially covered by a piece of paper. How many sides does N have?

(1) x + y = 45
(2) N is a regular polygon

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Re: In the figure above, polygon N has been partially covered by a piece  [#permalink]

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09 Mar 2018, 00:47
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Bunuel wrote:
In the figure above, polygon N has been partially covered by a piece of paper. How many sides does N have?

(1) x + y = 45
(2) N is a regular polygon

Bunuel Figure is missing. Please post the figure as well.
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Re: In the figure above, polygon N has been partially covered by a piece  [#permalink]

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09 Mar 2018, 00:50
GMATinsight wrote:
Bunuel wrote:
In the figure above, polygon N has been partially covered by a piece of paper. How many sides does N have?

(1) x + y = 45
(2) N is a regular polygon

Bunuel Figure is missing. Please post the figure as well.

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Re: In the figure above, polygon N has been partially covered by a piece  [#permalink]

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09 Mar 2018, 01:03
Bunuel wrote:

In the figure above, polygon N has been partially covered by a piece of paper. How many sides does N have?

(1) x + y = 45
(2) N is a regular polygon

Attachment:
Hidden_polygon_1.png

As per the figure, we see a triangle which has three angles x, y and an interior angle of the polygon

Each Exterior angle of a polygon = 360/n
Each Interior angle of a polygon = 180-360/n

where n = number of the sides in the regular polygon

Statement 1: x + y = 45

i.e. Third angle = 180 - (x+y) = 180-45 = 135
but we don't know if it's a regular polygon or not hence
NOT SUFFICIENT

Statement 2: N is a regular polygon
But we can't predict the number of sides unless we have measure one of the angles of the polygon hence
NOT SUFFICIENT

Combining the two statements
i.e. Third angle = 180 - (x+y) = 180-45 = 135

Each Interior angle of a polygon = 180-360/n
where n = number of the sides in the regular polygon

i.e. 135 = 180-360/n
i.e. n = 8 hence

SUFFICIENT

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Re: In the figure above, polygon N has been partially covered by a piece  [#permalink]

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14 Feb 2019, 09:41
Top Contributor
Bunuel wrote:

In the figure above, polygon N has been partially covered by a piece of paper. How many sides does N have?

(1) x + y = 45
(2) N is a regular polygon

Attachment:
Hidden_polygon_1.png

Target question: How many sides does N have?

Statement 1: x + y = 45
Since all angles in a triangle add to 180°, we know that the missing angle is 135°

There are plenty of polygons that have at least one angle measuring 135°. Here are two:

Case a:

In this case, the answer to the target question is polygon N has 3 sides

Case b:

In this case, the answer to the target question is polygon N has 4 sides

Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: N is a regular polygon
This definitely doesn't help (we have no idea what the measurement of each angle is)
Statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that ONE angle measures 135°
Statement 2 tells us that all of the angles are EQUAL (that's what "regular" means)
At this point, we have LOCKED IN the shape. That is, there is ONLY ONE regular polygon in which all of the angles are 135°
So, if we did some more work, we COULD determine the number of sides of N, which means we COULD answer the target question with certainty
The combined statements are SUFFICIENT

Aside: Here's how we'd determine the number of sides:
Useful rule: the sum of the angles in an n-sided polygon = (n - 2)(180º)
So, in a REGULAR n-gon, the measurement of EACH angle = (n - 2)(180º)/n

We can write: (n - 2)(180)/n = 135
Multiply both sides by n to get: (n - 2)(180) = 135n
Expand left side to get: 180n - 360 = 135n
Rearrange to get: 45n = 360
Solve: n = 360/45 = 8
So, polygon N has 8 sides

Cheers,
Brent
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Re: In the figure above, polygon N has been partially covered by a piece   [#permalink] 14 Feb 2019, 09:41
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