MHB Find the diameter of one circle

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The problem involves finding the diameter of one circle among five identical semicircles arranged in a specific configuration. The total gaps above the semicircles sum to 24, while the gaps below total 60, leading to a difference of 36 for the diameter of the semicircle. The discussion emphasizes solving the problem without algebra, although some participants acknowledge that their methods still involve algebraic reasoning. Various approaches and diagrams are shared to illustrate different methods of arriving at the solution. Ultimately, the diameter of one semicircle is confirmed to be 36.
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Five identical semicircles are arranged as shown. Find the diameter of one circle.
[TIKZ]
\draw (0,0) -- (16.5, 0);
\begin{scope}
\clip (0,0) rectangle (4.5,4.5);
\draw (2.25,0) circle(2.25);
\draw (0,0) -- (4.5,0);
\end{scope}
\begin{scope}
\clip (6,0) rectangle (10.5,4.5);
\draw (8.25,0) circle(2.25);
\draw (6,0) -- (10.5,0);
\end{scope}
\begin{scope}
\clip (12,0) rectangle (16.5,4.5);
\draw (14.25,0) circle(2.25);
\draw (12,0) -- (16.5,0);
\end{scope}
\begin{scope}
\clip (2.75,0) rectangle (7.25,-4.5);
\draw (5,0) circle(2.25);
\draw (2.75,0) -- (7.25,0);
\end{scope}
\begin{scope}
\clip (9.25,0) rectangle (13.75,-4.5);
\draw (11.5,0) circle(2.25);
\draw (9.25,0) -- (13.75,0);
\end{scope}
\draw [<->] (4.5, 0.5) -- (6, 0.5);
\draw [<->] (10.5, 0.5) -- (12, 0.5);
\draw [<->] (0, -0.5) -- (2.75, -0.5);
\draw [<->] (7.25, -0.5) -- (9.25, -0.5);
\draw [<->] (13.75, -0.5) -- (16.5, -0.5);
\coordinate[label=left:12] (A) at (5.5,0.8);
\coordinate[label=left:12] (B) at (11.5,0.8);
\coordinate[label=left:22] (C) at (1.6,-0.8);
\coordinate[label=left:16] (D) at (8.6,-0.8);
\coordinate[label=left:22] (D) at (15.6,-0.8);
[/TIKZ]

As this is a Singapore primary math problem, it is understandable that one can solve it without the use of algebra method (form an equation and solve the equation is what I mean by algebra method). I enjoyed this problem quite a bit, therefore I wanted to post it here to let others to try to solve it without the use of algebra method...:)
 
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I would say that the sums of the gaps are 24 (above) and 60 (below). As there is one more semicircle above, its diameter is equal to the difference 36.
 
Thanks castor28 for your reply!

But, I still count that as an algebra method, hehehe...I will let others have a chance to take a stab at it before I post the so called without-algebra solution. Please stay tuned! :)
 
Hi castor28!

I don't know where my head was when I made the previous reply (Tmi) , I'm so sorry!(Sadface) Your answer is spot on!

Here is a diagram to illustrate a slightly different approach than castors28's method:

[TIKZ]
\draw (0,0) -- (16.5, 0);
\begin{scope}
\clip (0,0) rectangle (4.5,4.5);
\draw (2.25,0) circle(2.25);
\draw (0,0) -- (4.5,0);
\end{scope}
\begin{scope}
\clip (6,0) rectangle (10.5,4.5);
\draw (8.25,0) circle(2.25);
\draw (6,0) -- (10.5,0);
\end{scope}
\begin{scope}
\clip (12,0) rectangle (16.5,4.5);
\draw (14.25,0) circle(2.25);
\draw (12,0) -- (16.5,0);
\end{scope}
\begin{scope}
\clip (0,0) rectangle (4.5,-4.5);
\draw (2.25,0) circle(2.25);
\draw (0,0) -- (4.5,0);
\end{scope}
\begin{scope}
\clip (12,0) rectangle (16.5,-4.5);
\draw (14.25,0) circle(2.25);
\draw (12,0) -- (16.5,0);
\end{scope}
\draw [<->] (4.5, 0.5) -- (6, 0.5);
\draw [<->] (10.5, 0.5) -- (12, 0.5);
\draw [<->] (4.5, -0.5) -- (6, -0.5);
\draw [<->] (7.25, -0.5) -- (9.25, -0.5);
\draw [<->] (10.5, -0.5) -- (12, -0.5);
\draw [<->] (6, -0.5) -- (7.25, -0.5);
\draw [<->] (9.25, -0.5) -- (10.5, -0.5);
\coordinate[label=left:12] (A) at (5.5,0.8);
\coordinate[label=left:12] (B) at (11.5,0.8);
\coordinate[label=left:12] (C) at (5.5,-0.8);
\coordinate[label=left:16] (D) at (8.6,-0.8);
\coordinate[label=left:12] (D) at (11.5,-0.8);
\coordinate[label=left:10] (E) at (6.9,-0.8);
\coordinate[label=left:10] (F) at (10.1,-0.8);
[/TIKZ]

$\therefore \text{diameter}=10+16+10=36$
 
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