0
June 23, 2020, 2:56 a.m.
All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon do two or more diagonals intersect?

All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?

Options are

1. 49
2. 65
3. 70
4. 96

quant
geometry
permutation-combination
cat
0
June 23, 2020, 3:14 a.m.

Option 1 which is 49 is the correct answer.
Let the number of intersections be x.
We know that x <= 8C= 70, as every  vertices on the octagon forms a quadrilateral with intersecting diagonals which is an intersection point. However, four diagonals intersect in the center, so we need to subtract 4C2 - 1 = 5. (70 - 5 = 65)
There are 8 type of three diagonals intersecting at the same point, so we need to subtract 8 x 2 = 16 (as we will count it once)
65 - 16 = 49
Another simple method using option

As we know maximum number of intersection is 70. there will be four diagonals intersect in the center, so we need to subtract 4C2 - 1 = 5. (70 - 5 = 65)Probably there will be more.
We have only 1 option which is less than 65.
i.e. 49

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numbers
cat-2019
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rc
set-theory
time-speed-distance
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