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?
- 10
- 25
- 58
- 90
- 91
Approach Solution (1)
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)
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)
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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