Consider a rectangle which has a diagonal of length 6. If \(P\) is a point on side \(AB\) such that \(\lvert\overline{AP}\rvert=\lvert\overline{AD}\rvert,\) and \( Q\) is a point on the extension of side \(AD\) such that \(\lvert\overline{AQ}\rvert=\lvert\overline{AB}\rvert,\) what is the area of the quadrilateral \(APCQ?\)

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In triangle \(ABC,\) \(AB = AC\) and \(\angle BAC = 100^\circ.\) If \(\overline{AB}\) is extended to \(D\) such that \(AD=BC,\) find \(\angle BCD\) (in degrees).

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\(ABC\) is an isosceles triangle with \(AC = BC\). Furthermore, \(D\) is a point on \(BC\) that bisects the angle at \(A\). If \(\angle B = 72^\circ\) and \(CD=1,\) then find length of \(BD\)(upto 3 decimal places).

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In a triangle with integer side lengths, one side is three times as long as a second side, and
the length of the third side is 17. What is the greatest possible perimeter of the triangle?

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