[Not-so-daily puzzle] Purple area \ stacker news ~science

pull down to refresh

545 sats \ 15 comments \ @south_korea_ln 13 Dec science

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)

view all related items

25 sats \ 6 replies \ @0xbitcoiner 13 Dec

I don't even know where to start! Isn't the 'center' of the polygon poorly represented? 😂

29 sats \ 5 replies \ @south_korea_ln OP 13 Dec

It's not the center... it's a point inside the polygon.

28 sats \ 4 replies \ @0xbitcoiner 13 Dec

joking! That's all I've managed to find out for now. When I have more time, I'll pick it up again.

0 sats \ 3 replies \ @frostdragon 13 Dec

Can we assume that the lines that go from 12:00 to 5:30 and 1:30 to 7:30 are straight lines?

0 sats \ 2 replies \ @frostdragon 13 Dec

8:00, not 7:30

0 sats \ 0 replies \ @south_korea_ln OP 14 Dec

#809252

0 sats \ 0 replies \ @south_korea_ln OP 14 Dec

It's a regular polygon, i.e. all the dots on the circle are equally spaced...

0 sats \ 7 replies \ @SimpleStacker 14 Dec

Can any of the lines from one vertex to another (passing through the middle point), be assumed to be straight?

0 sats \ 6 replies \ @south_korea_ln OP 14 Dec

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.

0 sats \ 5 replies \ @Scroogey 14 Dec

I guess the question is, are the green lines intersecting chords through one and the same point?

0 sats \ 4 replies \ @south_korea_ln OP 14 Dec

Oh, got it.

Yes. The answer is yes, those are straight lines. I was a bit too jetlagged when answering #808331 or #808835.

0 sats \ 3 replies \ @Scroogey 14 Dec

I think the area of the magenta region is

. Let me know if that's correct, and I'll explain.

563 sats \ 2 replies \ @Scroogey 14 Dec

If we look at the upper blue region:

The area is the sum of the components:

Adding the same (with

instead of

) for the lower blue region:

The apothem formula gives

Simple trig says

Those last four equations solve for

, or total area of polygon

As seen in @0xbitcoiner's drawing, the total area is the sum

Therefore,

0 sats \ 0 replies \ @0xbitcoiner 15 Dec

I'm not going to be the one checking those calculations! 😎👍

0 sats \ 0 replies \ @south_korea_ln OP 14 Dec

Correct, the area of the magenta region equals $10$.