You mentioned being concerned about the sectors being more spread out initially:
The part that is more spread out are the triangles though (the area part), NOT the line that makes up the circumference. Like, if they were spread out so that the circumference line was broken/no longer touching between the different triangles, then the length would end up being longer than the circumference. But the outer edge of the circle doesn’t end up being spread out in the rectangle - only the inner parts of the circle do.
BTW I think you’re not supposed to use pythagorean theorem for this cuz that’s introduced in next chapter.
Since E, F, G, and H are midpoints of the sides, and since the shape is a square, I figured out that each short segment on the perimeter (line AH, line AB, line BE, and so on, around the square) is length 10.
My first attack of the problem went something like: well the area of the whole square is 20 x 20 = 400. Each short segment is length 10. So then I was like, well, the area of triangle ABE is \frac{1}{2}(10 * 20) = 100 and same with triangle CDG. So I was gonna subtract those from the area of the square. But then I ran into the issue that there are two quadrilaterals (to the top left and bottom right of the shaded area) that I haven’t accounted for, so that doesn’t work.
My second attack was to try to creatively overcounting and working my way back. What I mean is, there are big triangles “around” the shaded square that we can calculate the area of = namely ABE, BCF, CDG, and DAH. The area of those is 100. But if you just add them all up, you get 400, which is the area of the whole square. The issue is that there are four smaller triangle segments that are getting double counted. Now if you could figure out the area of those, you could compensate for the overcounting and thus figure out the area of the shaded area. I was unable to come up with a way of doing that without pythagorean theorem though.
I didn’t “get” there hint, don’t know what they have in mind.
I initially took their hint as meaning to extend lines outside the square. That’s a common technique. I quickly got a solution by looking at it that way:
I’m not sure if you’re asking for help or hints or not.
I solved it before reading your thoughts, but I used Pythagorean theorem the first time, since that seemed most intuitive to me. After reading that they didn’t intend Pythagorean to be used in the solution, I figured out a different way to solve it without using Pythagorean. I’m not sure if you want hints or help or anything though, so I don’t want to give more details.
Oh my bad. Hints/suggestions/help welcome. I guess start with hints. I looked at Elliot’s suggestion but haven’t figured out how to connect it to solving the problem.
OK so the basic technique here is to look at shapes that use some but not all the lines. You did that already when you saw the ABE triangle. To see that triangle, you ignored the HD and BF lines. Similarly, you might notice a triangle with A, B and the point touching the black square near E. To see that medium sized triangle you ignore the HD line. There’s also the small triangle. And if you look at my big pic, you can find some larger triangles.
ok so do any of the larger triangles look useful? part of the way to see if they’re useful is by looking at how they can be broken up internally, so you can try to say the big shape equals several smaller shapes.
for example, if you find a large triangle that is the same area as the original square, you might be able to look at the components in it and know they add up to 400 area. that could give you a different set of components, besides the ones in the original square, that add to 400. so that’d help you figure it out.
and can you see any other shapes besides triangles?
I tried Anne’s hint. It is different from what I did, but I was able to find the area that way too (though I am not sure if the way that I did it was the way she intended).
Edit: actually, thinking about it, I am not sure if I solved it using Anne’s hint alone, or if I was taking advantage of things that I already knew from solving it myself and/or looking at Elliot’s image.
It would have the same area as original square (cuz the left side is 20 and the bottom is 40, so (20 * 40)/2 = 400). I’m getting stuck re: what to actually do with that info though, how to use it.
What are the shapes within this big triangle? And what are the shapes within the original square? Are they different? If they are different that’s useful.
To clarify what I mean, the original square contains:
