How Many Diagonals Does A Polygon With 18 Sides Have If Three Of Its GMAT Problem Solving

Question: How many diagonals does a polygon with 18 sides have if three of its vertices, which are adjacent to each other, do not send any diagonals?

  1. 10
  2. 25
  3. 58
  4. 90
  5. 91

Approach Solution (1)

Total number of diagonals with no restriction= (18 * (18 - 3)) / 2 = 135
Restriction: tThree of its vertices, which are adjacent to each other, do not send any diagonals:
Each vertex sends 15 diagonals (15 = 18 - 1(the vertice) - 2(adjacent vertices)). Thus 3 vertices DO NOT send 15*3 = 45 diagonals.
Then, out of these 3 vertices, 2 are not adjacent to each other, thus they share 1 diagonal (double counted in 45).
Total number of diagonals: 135 - (45-1) = 91

Correct option: E

Approach Solution (2)

Polygon has 18 vertices and 3 vertices do not send a diagonal. This means the number of vertices which will send diagonals is 15.
Total number of lines possible with 15 vertices = C(15,2) = 105
[Total number of lines = Number of sides + Number of possible diagonals ]
Polygon Number of vertices Polygon Number of sides
18 vertices 18 sides
15 vertices 14 sides {since 3 vertices do not send a diagonal they cannot form a side. So, the count will be 14 and not 15}
Total number of diagonals if 3 vertices do not send any diagonal = 105 – 14 = 91

Correct option: E

Approach Solution (3)

For any n-sided polygon, the number of diagonals = n*(n - 3)/2
There are 18 sides or 18 vertices of which 3 vertices cannot send any diagonal
Therefore, number of diagonals = (15 * 12)/2 = 90
Out of 3 vertices, there are 2 vertices that are not adjacent to each other, so they will share one 1 diagonal
So, total number of diagonal = 90 + 1 = 91

Correct option: E

“How many diagonals does a polygon with 18 sides have if three of its vertices, which are adjacent to each other, do not send any diagonals?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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