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# A polygon has 12 edges. How many different diagonals does it have?

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A polygon has 12 edges. How many different diagonals does it have? [#permalink]

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20 Sep 2017, 14:29
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A polygon has 12 edges. How many different diagonals does it have?

(A) 54
(B) 66
(C) 108
(D) 132
(E) 144
[Reveal] Spoiler: OA

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A polygon has 12 edges. How many different diagonals does it have? [#permalink]

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20 Sep 2017, 14:49
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Gnpth wrote:
A polygon has 12 edges. How many different diagonals does it have?

(A) 54
(B) 66
(C) 108
(D) 132
(E) 144

The number of diagonals for a polygon, where n is the number of sides, is found by the formula

$$\frac{(n)(n - 3)}{2}$$

$$\frac{(12)(9)}{2}$$ = 54

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Re: A polygon has 12 edges. How many different diagonals does it have? [#permalink]

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20 Sep 2017, 14:59
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Gnpth wrote:
A polygon has 12 edges. How many different diagonals does it have?

(A) 54
(B) 66
(C) 108
(D) 132
(E) 144

The polygon in question has 12 vertices.

Let's focus on ONE VERTEX, which we'll call vertex A
How many diagonal can be drawn from vertex A?
We'll there are 11 other vertices to connect to, HOWEVER the vertices on either side of vertex A are out of contention, since joining two adjacent points will not create a diagonal.
So, we can create a diagonal with vertex A by connecting to any of the 9 eligible vertices.

Using the same logic, we can conclude that we can create a diagonal with vertex B by connecting to any of the 9 eligible vertices.

And we can create a diagonal with vertex C by connecting to any of the 9 eligible vertices.

There are 12 vertices, so the TOTAL number of diagonals = (12)(9) = 108
HOWEVER, this is not the final answer, since we have counted each diagonal TWICE.
For example, in the first step, we counted the diagonal created by joining vertex A with vertex B, and in the second step, we counted the diagonal created by joining vertex B with vertex A

To account for this duplication, we must take 108 and divide it by 2 to get: 54

[Reveal] Spoiler:
A

Cheers,
Brent
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Re: A polygon has 12 edges. How many different diagonals does it have?   [#permalink] 20 Sep 2017, 14:59
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