Consider a convex polygon such that:
- Each of its sides correspond to the square root of a positive integer.
- The polygon can be inscribed in an unit circle.
Determine the total number of polygons that simultaneously satisfy the 2 properties mentioned above.
Note: Polygons that are rotations and reflections of each other are considered the same.
The integer whose square root is one side cannot be larger than 3, as sqrt(4) ia already 2, which is the diameter of the inscriping circle.
For each of the squares of the sides, 1, 2 and 3, follows the number of degrees subtended at the center of the circle by the side:
1 60
2 90
3 120
(based on twice the arcsin of half the square root)
The subtended angles must add up to 360°.
ways of arranging
120 + 120 + 120 1 `
120 + 120 + 60 + 60 2 (alternating vs together)
120 + 90 + 90 + 60 2 (60 opposite or adjacent 120)
120 + 60 + 60 + 60 + 60 1
90 + 90 + 90 + 90 1
90 + 90 + 60 + 60 + 60 2 (90's adjacent or separated)
60 + 60 + 60 + 60 + 60 + 60 1
---
10
So there are 10 such polygons.
|
|
Posted by Charlie
on 2023-04-29 08:14:14 |
solution
