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Catriona Shearer
The red line, of length 2, is perpendicular to the bases of the three semicircles. What’s the total shaded area?
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Bilal Oct 23
Replying to @Cshearer41
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Catriona Shearer Oct 23
Replying to @Gogoljecco
So close! But don’t forget they’re semicircles.
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Professor Smudge Oct 23
Replying to @Cshearer41
This is an intriguing question. You'd think the area might depend on the location of the red line. But if it doesn't, then the left-hand figure suggests the area is half-pi-2squared minus pi-1squared = pi
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Catriona Shearer Oct 23
Replying to @ProfSmudge
That was my thought too - how can it be enough information?! I’m waiting for someone to come up with a nice geogebra animation showing all the possible ways it can be drawn 😁
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Dr M Thornber Oct 23
Replying to @Cshearer41
It’s a pity we don’t teach the intersecting chords theorem any more!
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Catriona Shearer Oct 23
Replying to @DrMThornber
I learned the intersecting chords theorem via twitter a couple of months ago. Now I look for any excuse to use it 😁
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Swaraj Kumar Oct 23
Checked other responses, I always make the silly mistakes.
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Catriona Shearer Oct 23
That one seems to have caught quite a few people out today!
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kyle Oct 23
It’s impossible.
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Andreas Steiger Oct 23
Replying to @Cshearer41 @ProfSmudge
Geogebra here:
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Catriona Shearer Oct 23
Amazingly, it is possible. The length of that line is enough to calculate the whole area, although it took me a while to convince myself it would work!
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Catriona Shearer Oct 23
Thank you! 😁
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Alex Cutbill Oct 23
Replying to @Cshearer41
For those who are interested, the shaded region is an 'arbelos':
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Catriona Shearer Oct 23
Replying to @intersectarian
Thanks! I was sure there would be a name for it - it seems far too nice a result not to have one - but it's the kind of thing that's quite hard to search for when you start with a picture. There are some other interesting results on that page, too.
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Ally-mai Oct 23
If you are allowed to assume that: The radius of the larger semicircle must be 2, which means its area is 2π. The radi of the smaller semicircles must be 1, which means that their areas are π/2 each, so their areas add to π. Then the shaded area must be 2π-π=π.
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Catriona Shearer Oct 23
Nice use of an assumption to simplify it 😄 but now, can you convince yourself that the area won't change as the smaller semicircles change size?
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Ally-mai Oct 23
Do I need to treat it as a more general thing?
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david Oct 23
Replying to @Cshearer41
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Catriona Shearer Oct 23
The question kind of implied that the area would be there same whatever, so I’d say you were just taking an efficient short cut! But it’s interesting to work out why the area doesn’t depend on the relative sizes of the semicircles.
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