In a polygon with 9 equal sides and 9 equal angles

Question

In a polygon with 9 equal sides and 9 equal angles, how many different diagonals can be drawn ?

(A) 18
(B) 27
(C) 36
(D) 54
(E) 72

The OA should be (B), however I'm not sure 100% about it; this exercise belong to a set of problem given to me by a friend for training.

get2aditya · Accepted Answer

To find the number of diagonals in a polygon with n sides, use the following formula: n (n-3) / 2. 9*6/2 = 27. Here’s where the diagonal formula comes from and why it works. Each diagonal connects one point to another point in the polygon that isn’t its next-door neighbor. In an n-sided polygon, you have n starting points for diagonals. And each diagonal can go to (n – 3) ending points because a diagonal can’t end at its own starting point or at either of the two neighboring points. So the first step is to multiply n by (n – 3). Then, because each diagonal’s ending point can be used as a starting point as well, the product n(n – 3) counts each diagonal twice. That’s why you divide by 2. https://www.dummies.com/how-to/content/h ... lygon.html

Bunuel · Answer

The # of diagonals in  n  sided polygon equals to  C^2_n-n=\frac{n(n-3)}{2} :  C^2_n  choose any two vertices out of  n  points to connect  minus   n  sides, which won't be diagonals.

So, # of diagonals in 9 sided polygon is  \frac{n(n-3)}{2}=27 .

Answer: B.

Bunuel · Answer

Similar question to practice: how-many-diagonals-does-a-polygon-with-21-sides-have-if-one-101540.html Hope it helps

CrackverbalGMAT · Answer

At the outset, I doubt whether this could qualify as a 700- level question. Because, there’s only one concept that is being tested here and that is the number of a diagonals of a polygon. If you are aware of this concept, there’s not much left in this question from that point on.

For a polygon with n sides, the number of line segments that belong to that polygon is given by  n_C_2 . This is because, an n-sided polygon has n vertices of which any 2 need to be selected to form a line segment belonging to the polygon. 
‘n’ of these line segments need to be removed since these represent the sides of the polygon. Once this is done, we will be left with ( n_C_2  - n) line segments that represent the diagonals of that polygon.

The number of diagonals of a n-sided polygon =  n_C_2  – n .

Note that this formula is not restricted to regular polygons. It can be used for both regular and irregular polygons.

Side note:  A regular polygon is a polygon in which all the sides are equal and all the interior angles are equal , as is the case with the polygon defined in this question.

The value of n in this question is 9 since the question says the polygon has 9 sides. Therefore, number of diagonals of this polygon =  9_C_2  – 9 = (9*8/2) – 9 = 36 – 9 = 27.
The correct answer option is B.

Note that  n_C_2  =  \frac{n(n-1) }{ 2} .

Hope that helps!

In a polygon with 9 equal sides and 9 equal angles

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In a polygon with 9 equal sides and 9 equal angles

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