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Math Expert V
Joined: 02 Sep 2009
Posts: 57290
A star-polygon is drawn on a clock face by drawing a chord from each  [#permalink]

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1
4 00:00

Difficulty:   65% (hard)

Question Stats: 53% (02:37) correct 47% (03:00) wrong based on 32 sessions

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A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star polygon?

A. 20
B. 24
C. 30
D. 36
E. 60

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A star-polygon is drawn on a clock face by drawing a chord from each  [#permalink]

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Bunuel wrote:
A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star polygon?

A. 20
B. 24
C. 30
D. 36
E. 60

Upon drawing figure we would realize that the figure been formed is a 12 angled star shape figure which starts at 12 and goes around clock at every 5th number and ends at 12 , in short we get 12 sides of figure ; so since total angle sum 360 so each angle = 360/12 ; 30
IMO C
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CrackVerbal Quant Expert S
Joined: 12 Apr 2019
Posts: 230
Re: A star-polygon is drawn on a clock face by drawing a chord from each  [#permalink]

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2
Now, although at first, this question can look difficult and overwhelming, as you plot the figure, you will observe that there are more than 1 ways of answering this question.

This might not exactly be a GMAT type of question because Star polygons are non-convex polygons and GMAT does not test you on non-convex polygons. However, this is definitely a good question to test out your knowledge of circle concepts.

From the description of the polygon given in the question, this looks like a self-intersecting regular star polygon, since the length of each side is the same.
A self-intersecting regular star polygon is an equiangular polygon. So, all the angles of this polygon will be equal.

Let us draw the polygon now:

Attachment: 2nd Aug 2019 - Reply 1 - 2.JPG [ 28.27 KiB | Viewed 265 times ]

Looks brilliant, isn't it?

As mentioned by Archit in his reply, we see that the polygon obtained is a 12 sided polygon. Since the method involving the sum of the interior angles is already discussed, let’s look at another way of solving this question.

This method is based on the angle made by the arcs touched by the sides of the polygon. To make things clearer, here’s another diagram, showing only the first 2 sides of the polygon.

Attachment: 2nd Aug 2019 - Reply 1 - 1.JPG [ 53.65 KiB | Viewed 267 times ]

Clearly, the angle made by arc 10-11-12 at the centre is 60 degrees, since each arc on a clock corresponds to an angle of 30 degrees. Since the arc 10-11-12 makes 60 degree at the centre, it will make exactly half i.e. 30 degrees at any point on the circumference.

This is precisely the angle that we were looking for. So, the interior angle of this polygon is 30 degrees.
The correct answer option is C.

Hope this helps!
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Joined: 17 Jul 2017
Posts: 119
A star-polygon is drawn on a clock face by drawing a chord from each  [#permalink]

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ArvindCrackVerbal wrote:
Now, although at first, this question can look difficult and overwhelming, as you plot the figure, you will observe that there are more than 1 ways of answering this question.

This might not exactly be a GMAT type of question because Star polygons are non-convex polygons and GMAT does not test you on non-convex polygons. However, this is definitely a good question to test out your knowledge of circle concepts.

From the description of the polygon given in the question, this looks like a self-intersecting regular star polygon, since the length of each side is the same.
A self-intersecting regular star polygon is an equiangular polygon. So, all the angles of this polygon will be equal.

Let us draw the polygon now:

Attachment:
2nd Aug 2019 - Reply 1 - 2.JPG

Looks brilliant, isn't it?
ArvindCrackVerbal
please clear mey doubt
total sum of angles =(n-2)*180
s0 12-2*180=1800
now each interior angle=1800/12=150
where m i going wrong

As mentioned by Archit in his reply, we see that the polygon obtained is a 12 sided polygon. Since the method involving the sum of the interior angles is already discussed, let’s look at another way of solving this question.

This method is based on the angle made by the arcs touched by the sides of the polygon. To make things clearer, here’s another diagram, showing only the first 2 sides of the polygon.

Attachment:
2nd Aug 2019 - Reply 1 - 1.JPG

Clearly, the angle made by arc 10-11-12 at the centre is 60 degrees, since each arc on a clock corresponds to an angle of 30 degrees. Since the arc 10-11-12 makes 60 degree at the centre, it will make exactly half i.e. 30 degrees at any point on the circumference.

This is precisely the angle that we were looking for. So, the interior angle of this polygon is 30 degrees.
The correct answer option is C.

Hope this helps!

ArvindCrackVerbal
please clear mey doubt
total sum of angles =(n-2)*180
s0 12-2*180=1800
now each interior angle=1800/12=150
where m i going wrong A star-polygon is drawn on a clock face by drawing a chord from each   [#permalink] 02 Aug 2019, 02:31
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