Let two square mats be placed with one corner of one in contact with one corner of another. It is tempting to surround the in between place for a game, a deity, a yagya or just leave vacant by placing a square mat at $PQ$.
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| Sitting Space 1 |
The sitting space $S$ provided by the third mat as shown in Figure \ref{sit-1} is not as much as provided by the earlier mats. If we increase the angle $\theta$ in between the two mats we started with, the space $S$ is seen to increase.
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| Sitting Space 2 |
\[ S < A,\ \ S > A. \]
From the situations $S < A$, can it suddenly turn to situations $S > A$?
Must it not pass through the situation $S = A$?
What is the corresponding value of $\theta$?
Is it not interesting to experiment that as we increase $\theta$ further, up goes $S$ (above $A$). Continue the experiment by increasing $\theta$ and observing the increase in $S$. What is the maximum possible value of $\theta$, and what is the corresponding value of $S$, in terms of $A$?
It is interesting to discover that you cross the situations $S > 2A$, $ S > 3A,\, \ldots$.
What happens as long as $\theta$ is acute; and what happens when it turns obtuse? Mark the turn from situations $S < 2A$ to $S > 2A$; can we characterise them in terms of angle $\theta$ (acute and obtuse values of $\theta$ ). In between must be the stage $S = 2A$. What is $\theta$ then? Formulate your discovery as a theorem.
Let us repeat the experiment with the hands of a clock, attaching stiff paper to hands making them sides of two squares with spaces $A$ and $B$ in between varying angle $\theta$. Discuss situations for acute and obtuse values of $\theta$.
Can situations $ S < A+B$ suddenly turn to $S > A+B$? When is $S = A+B$? Formulate your finding in the form of a theorem. Investigate whether your theorem remains true in case, the sitting mats we start with are semicircular or triangular (equilateral) instead of being square.
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