The distances from a certain point inside a regular hexagon to three of its consecutive vertices are equal to 1, 1 and 2, respectively. Find the length of this hexagon's side.
If the consecutive vertices are A, B, C then the point must be equidistant from (wlog) A and B. It could not be equidistant from A and C and still have the distance to B be double.
Say A is at (0,0), B is at (s,0), and C is at (3s/2, √3*s/2).
Then the point is somewhere on the vertical line x = s/2.
So point P has coordinates (s/2, h*s)
Graphical solution in Desmos:
https://www.desmos.com/calculator/lkjdi9z1c6
s is approx 1.74 (? √3)
h is approx 0.286
s*h is approx 0.5
putting exact values into Desmos
https://www.desmos.com/calculator/zlkznvnhii
Conclusion: the side length of the hexagon is √3.
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Posted by Larry
on 2024-06-03 10:45:02 |
Graphical assistance
