A certain game board is in the shape of a non-convex

Question

A certain game board is in the shape of a non-convex polygon, with spokes that extend from each vertex to the center of the board. If each spoke is 8 inches long, and spokes are used nowhere else on the board, what is the sum of the interior angles of the polygon?

(1) The sum of the exterior angles of the polygon is 360º.

(2) The sum of the exterior angles is equal to five times the total length of all of the spokes used.

mkl_in_2001 · Answer

Got B.

from 2
 The sum of the exterior angles is equal to five times the total length of all of the spokes used. 
Let say Y= # of sides and Y sum of interior angles
for any type of polygon the interior angle and exterior angle always add to 180Â°.

5*8*N + Y = 180*N
=>40N+Y = 180 N
and 
Y = (N-2)*180

N = 9 and
Y =(9-2)*180= 7*180
So B
whats the OA?

mkl_in_2001 · Answer

40N+Y = 180 N ------ sum of exterior angles + sum of interior angles
Y = (N-2)*180 --------sum of interior angles

From those two eqations 
U will get N= 9

anything wrong?

Hi mm007,
 Do u have OA for this?

nitinneha · Answer

what is meant by non-convex polygon here?

mkl_in_2001 · Answer

non-convex polygon means at least of the interior angles >180 deg.

mm007 · Answer

OA is B...

Here is the OE:

The formula for the sum of the interior angles of a non-convex polygon is (n â€“ 2)(180), where n represents the number of sides. To find the sum of the interior angles of the polygon then, we need to know the number of sides. We can therefore rephrase the question:

How many sides does the game board have?

(1) INSUFFICIENT: It tells us nothing about the number of sides. The sum of the exterior angles for any non-convex polygon is 360.

(2) SUFFICIENT: The sum of the exterior angles = 5 Ã— length of each spoke Ã— number of spokes. 

360 = 5(8)(x)
360 = 40x
9 = x

The game board has nine sides. The sum of its interior angles is (9 â€“ 2)(180) = 1260.

The correct answer is B.

mkl_in_2001 · Answer

Got B too but in different way ; 

(2) SUFFICIENT: The sum of the exterior angles = 5 Ã— length of each spoke Ã— number of spokes. 

360 = 5(8)(x)
how did u get this 360 as sum of the exterior angles; (2) does not say that
unless you deduce it .

bhaarat · Answer

In my opinion the answer should be C. We don't know what is the sum of exterior angles. We have to use this information from the first statement,then only we can set up this equation
360=5(8x)
that is sufficient to know the number of sides and hence the sum of interior angles.

Convex polygon is a polygon inwhich each interior angle has a measure of less than 180.

wizard · Answer

The Exterior Angles of a Polygon always equal 360 degrees, no matter the number of sides the polygon has.

JusTLucK04 · Answer

How can the sum of all exterior angles of a non convex polygon be 360..I take a polygon with n sides and the sides are so acute tat exterior angle is close to 360..Imagine something like the shape of a star just with more side..

rohitmanglik · Answer

Sum of a convex polygon is always 360.

An exterior angle of a polygon is an angle that forms a linear pair with one of the angles of the polygon.

Two exterior angles can be formed at each vertex of a polygon. The exterior angle is formed by one side of the polygon and the extension of the adjacent side. For the hexagon shown at the left, <1 and <2 are exterior angles for that vertex. Be careful, as <3 is NOT an exterior angle.

Source: https://www.regentsprep.org/

JusTLucK04

How can the sum of all exterior angles of a non convex polygon be 360..I take a polygon with n sides and the sides are so acute tat exterior angle is close to 360..Imagine something like the shape of a star just with more side..

rohitmanglik · Answer

Good Question, In case of a concave (non-convex) polygon, you subtract exterior angle from the equation (of sum). E.g. check out attached image as an illustration. Hope it helps

kham71 · Answer

This is a lot easier than it sounds.

What is the sum of the interior angles of the polygon? ==> What is (n-2)*180? ==> What is n?

Statement 1: Supplies a known geometric fact that adds no new information. The sum of exterior angles of a polygon is always 360. This gives no information about n, insuff.

Statement 2: 5*n*8 = sum of ext angles ==> 40*n = 360 ==> n = 9. Suff.

Answer is B

shringi87 · Answer

Bunuel  - is this question in gmat gambit ?? and how is it deduced with n=5. Can u pls look into this and proivde your say

Thanks !!

ashisplb · Answer

Perhaps the question is wrong. Non convex polygons can't have center point from which all vertex will be equidistant

Posted from my mobile device

ccooley · Answer

I suspect this problem was intended to read either 'convex' or 'non-concave', based on the official solution in the Manhattan Prep resources. The point made above is correct as well - a concave polygon won't have a center equidistant from all vertices.

kornn · Answer

Dear Bunuel VeritasKarishma chetan2u GMATGuruNY IanStewart DmitryFarber RonPurewal ccooley MathRevolution,

Q1. Is statement 1 true for both convex and non-convex polygon?
I think it should apply ONLY to convex polygon. See here: https://www.khanacademy.org/math/geomet ... ex-polygon

Q2. For statement 2, a non-convex polygon WON'T have a center equidistant from all vertices. How can we possibly solve this problem?

IanStewart · Answer

This entire question is a mess. You do not need to know what the words "convex" or "concave" mean on the GMAT, nor do you need to know anything about "exterior angles". There also seems to be an error in the stem when they use the word "non-convex". And Statement 2 is nonsensical, mathematically. You can't compare a length with an angle. They are in different units. It's like if I told you "the length of time it will take me to walk to your house is equal to the distance from here to your house". That sentence is meaningless - in what units are we measuring time and distance? You can compare two angles, or two distances, or two times, or two numbers. You can't compare a distance and an angle.

There is no reason to study this question.

A certain game board is in the shape of a non-convex

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