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Regular Polygons

With equal angles and equal side lengths, what more could you want from a polygon?

Regular Polygons - Decomposition into Triangles

         

Consider a rectangle with diagonals as shown in the above diagram. If the area of \(\triangle BCE\) is \(32,\) what is the area of \(\triangle ABE?\)

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The above diagram is a regular pentagram. Let \(\lvert\triangle TRS\rvert\) denote the area of \(\triangle TRS,\) and \(\lvert\square PQRS\rvert\) the area of \(\square PQRS.\) If \(\lvert\triangle ABC\rvert=52,\) what is the area of the shaded region \[\lvert\square PQRS\rvert - \lvert\triangle TRS\rvert?\]

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In the above diagram, \(\square ABCD\) is a square and \(\overline{DE}\) is perpendicular to one of its diagonals \(\overline{AC}.\) If the length of \(\overline{DE}\) is \(21.\) what is the length of \(\overline{AC}?\)

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In the above diagram, figure \(ABCDE\) is a regular pentagon centered at point \(F\). Find the area of quadrilateral \(CDEF\) when the area of triangle \(ABF\) is \(56.\)

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In \(\triangle ABC\) above, \(L,M\) and \(N\) are the midpoints of \(\overline {AB}, \overline {BC}\) and \(\overline {AC},\) respectively. If the area of \(\triangle LMN\) is \(54,\) what is the area of \(\triangle ABC?\)

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