3 squares in semicircle

[nanase]

I assume if little square side is a, then the big square is 2a (double the size) is it correct?
after that I use pythagoras theorem to find the diagonal of big square (which is the radius)
then the area of semi circle is 1/2 . pi . ((root 5) a)squared = 5/2.pi a squared (is this the correct answer?)

 

[Dr.Peterson]

I assume if little square side is a, then the big square is 2a (double the size) is it correct?

This turns out to be correct, but I wouldn't say it is so obvious that you can assume it.

In fact, your answer is correct. If you can prove your assumption to be true, then you are finished.

 

[nanase]

This turns out to be correct, but I wouldn't say it is so obvious that you can assume it.

In fact, your answer is correct. If you can prove your assumption to be true, then you are finished.

Thanks Dr. Peterson, but can you elaborate on the reasoning why the big square is double the length of small square from geometrical point of view? I'm interested so much and feels my answer is not fully right if I am still "assuming"

 

[Dr.Peterson]

Thanks Dr. Peterson, but can you elaborate on the reasoning why the big square is double the length of small square from geometrical point of view? I'm interested so much and feels my answer is not fully right if I am still "assuming"

I'm asking you to explain why you think it is true. There are some good reasons to do so. My own approach to the problem took half the side of the large square as a second variable, rather than assuming anything about it, but I think you can make a good case directly, and I want to see your thinking. (Unless you just assumed it because it looks that way in the picture!)

 

[nanase]

I'm asking you to explain why you think it is true. There are some good reasons to do so. My own approach to the problem took half the side of the large square as a second variable, rather than assuming anything about it, but I think you can make a good case directly, and I want to see your thinking. (Unless you just assumed it because it looks that way in the picture!)

Yes at first I assumed it is double because of the diagram it self (looks twice the length haha!) but after your previous post here, I am thinking that if I draw a line from the centre of the circle to the top right of big square, I will form a triangle. And this will be a congruent triangle if I draw a line from the same centre to the top right of small square. (is this reasoning correct)?

can I see your approach that is using a second variable?

 

[Dr.Peterson]

Yes at first I assumed it is double because of the diagram it self (looks twice the length haha!) but after your previous post here, I am thinking that if I draw a line from the centre of the circle to the top right of big square, I will form a triangle. And this will be a congruent triangle if I draw a line from the same centre to the top right of small square. (is this reasoning correct)?

I don't see anything there to prove that the triangles are congruent:

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What you can see is that if b=a, then the triangles are congruent, so that this is a possible solution. It doesn't prove that it's the only possible solution.

What I did was to write two equations using the Pythagorean theorem, and solve the resulting (nonlinear) system (with r as a constant). You'll find that b=a is the only solution with both variables being positive.