An "interior segment" of a polygon is defined as any line segment that

Question

An "interior segment" of a polygon is defined as any line segment that can be drawn between two points of a polygon except for the sides of the polygon itself. If polygon P is a regular octagon, how many interior segments does P have?

(A) 20
(B) 28
(C) 36
(D) 48
(E) 56

(A) 20
(B) 28
(C) 36
(D) 48
(E) 56

ganand · Accepted Answer

Hi rishit1080,

In this question, we have to find the number of diagonals of an octagon.

The number of diagonals of any  n -sided polygon is =  \frac{n(n-3)}{2}

In this case, n = 8. Number of diagonals = 8*5/2 = 20.

Alternate Method:

Please refer the attached diagram. From each vertex, we can draw five lines. For example, from vertex A we can draw AC, AD, AE, AF, and AG.

Total number of lines = 8*5 = 40 lines. Please note that each line we are counting twice, so we have to divide it by 2.

Hence, total number of unique lines = 40/2 = 20.

Hope this helps.

rishit1080 · Answer

Tough for sub 600 as this question makes you draw an octagon and show the pattern so you don't get it wrong. Got A, but took time to show the different routes. 5 for first point, 5 for second point, 4 for third point, 3 for fourth point, 2 for fifth point, 1 for sixth point, and 0 for seventh and eighth points. Add them up to get 20.

Abhishek009 · Answer

With ganand regular-octagon-with-diagonals-from-a-vertex.jpg There are 8 points A, B,C, D, E, F, G & H From each point say point A , there can be 5 Diagonals as - 1. AC 2. AD 3. AE 4. AF 5. AG Now, Imagine Diagonals from point C , one of them will be AC , which is counted twice, one from Point A and the other from Point C... Total there will be 8*5 Diagonals = 40 Diagonals Due to counting Diagonals twice, actual no of Diagonals will be 40/2 = 20 diagonals... Thus, answer must be (A) 20

umg · Answer

Octagon has 8 Vertices.

So, the easiest way to do this one is by Combination. To draw a line, we need to pick 2 points. Therefore,

Number of ways to select 2 points out of 8 = 8C2

Combination because order of picking points is not important.

8 lines out of these will be the sides of the octagon.

Therefore, Total Number of "interior segment" or Diagonals = 8C2 - 8 =  \frac{8!}{(2! * 6!)} - 8  = 20

P.S. This is also the derivation of general formula for finding the Number of Diagonals of a closed polygon. 
In other words, Formula for finding the Number of Diagonals of a closed polygon is  nC2 - n

Yogananda · Answer

Here, we need to find the diagonals which connects 2 points of the octagan.
Formula is : n*(n-3)/2
8*(5)/2 = 20.

Hence, A.

Hope it helps.

An "interior segment" of a polygon is defined as any line segment that

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An "interior segment" of a polygon is defined as any line segment that

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