If the number of diagonals of polygon is 12 more than number of its si

Question

If the number of diagonals of polygon is 12 more than number of its sides, what is the number of sides of polygon ?

a)3

b)5

c)6

d)8

e)None

shrik · Answer

Let number of sides of polygon be N
 And diagonal be D
D=12+N

Remember number of diagonals of polygon is given by n(n-3)/2

N(N-3)/2 = 12+N

N^2-3N =24+2N

N^2-5N-24

N =8 OR N =-3

-3 IS NOT ACCEPTABLE SO N=8

Posted from my mobile device

BrentGMATPrepNow · Answer

A polygon with n sides will also have n vertices,

So, let n = the number of vertices of the polygon 
Each individual vertex can have n - 3 diagonals. Here's why: 
First, the vertex cannot create a diagonal when connected to itself. 
Second, the vertex cannot create a diagonal with the two points on either side of it, since that line will not be a diagonal. 
So, in a polygon with n vertices, each vertex will have n-3 diagonals connected to it.

So, the total number of diagonals = (n)(n - 3)
STOP! We made a mistake here. 
We must recognize that we counted every diagonal TWICE. 
For example, if the line joining vertex A and vertex F is a diagonal, we count that diagonal once when counting the (n-3) diagonals at vertex A,  AND  we count that diagonal once when counting the (n-3) diagonals at vertex F. 
To account for this duplication, we'll divide the number of diagonal by 2.

So, the   TOTAL number of diagonals = (n)(n - 3)/2

If the number of diagonals of polygon is 12 more than number of its sides, what is the number of sides of polygon ?  
We can write: (number of diagonals of polygon) = (number of its sides) + 12
In other words: (n)(n - 3)/2 = n + 12
Multiply both sides by 2 to get: (n)(n - 3) = 2n + 24
Expand: n² - 3n = 2n + 24
Subtract 2n from both sides: n² - 5n = 24
Subtract 24 from both sides: n² - 5n - 24 = 0
Factor: (n - 8)(n + 3) = 0
So, EITHER n = 8 OR n = -3

Since n must be POSITIVE, we know that n = 8

Answer: D

Cheers,
Brent

Kinshook · Answer

Asked: If the number of diagonals of polygon is 12 more than number of its sides, what is the number of sides of polygon ?

Let the number of sides of polygon = n

Number of diagonals = nC2 - n = n(n-1)/2 - n = n(n-3)/2 = 12 + n
n^2 - 3n = 24 + 2n
n^2 - 5n - 24 = 0
n^2 - 8n + 3n - 24 = 0
(n+3)(n-8) = 0
n = 8
The number of sides of polygon = n = 8

IMO D

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If the number of diagonals of polygon is 12 more than number of its si

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If the number of diagonals of polygon is 12 more than number of its si

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards