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Learn about packing shapes neatly into others, and the resulting geometric properties.

In the above diagram, square \(ABCD\) is inscribed in circle \(O,\) which is in turn inscribed in square \(A'B'C'D'.\) If the length of \(\overline{A'D'}\) is \(\lvert\overline{A'D'}\rvert=8,\) what is \(\lvert\overline{AD}\rvert?\)

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The area of the square inscribed in circle \(O\) in the above diagram is \(484.\) What is the area of the shaded region?

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The area of the circle in the above diagram is \(64\pi.\) What is the area of the inscribed square \(ABCD?\)

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In the above diagram, \(\square ABCD\) is a square inscribed in circle \(O,\) and \(\overline{ON}\) is perpendicular to \(\overline{BC}.\) If the length of \(\overline{MN}\) is \(2\sqrt{2}-2,\) what is the length of \(\overline{BM}?\)

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For which of the following quadrilaterals does a circumscribing circle always exist?

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