In a regular 24-sided polygon, lines are drawn from specific vertices to a point inside the polygon. If the total area of the red regions is 47 and the total area of the blue regions is 38, calculate the area of the magenta region.
Previous iteration: #801268 (unsolved, even with a 5000 sat bounty)
Can any of the lines from one vertex to another (passing through the middle point), be assumed to be straight?
If the line passes through the center point, yes.
If the line goes through any other point, it'll have a kink, as seen in the figure.
Edit: let me know if I misunderstood your question.
I guess the question is, are the green lines intersecting chords through one and the same point?
Oh, got it.
Yes. The answer is yes, those are straight lines. I was a bit too jetlagged when answering #808331 or #808835.
deleted by author
If we look at the upper blue region:
The area is the sum of the components:
Adding the same (with h instead of g) for the lower blue region:
The apothem formula gives
Simple trig says
Those last four equations solve for s≈1.82625, or total area of polygon A=152.
As seen in @0xbitcoiner's drawing, the total area is the sum
Therefore, x=10
I'm not going to be the one checking those calculations! 😎👍
Correct, the area of the magenta region equals $10$.
I don't even know where to start! Isn't the 'center' of the polygon poorly represented? 😂
It's not the center... it's a point inside the polygon.
joking! That's all I've managed to find out for now. When I have more time, I'll pick it up again.
Can we assume that the lines that go from 12:00 to 5:30 and 1:30 to 7:30 are straight lines?
8:00, not 7:30
#809252
It's a regular polygon, i.e. all the dots on the circle are equally spaced...