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Regular Polygons

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 \(\triangle ABE?\)

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 \[\lvert\square PQRS\rvert - \lvert\triangle TRS\rvert?\]

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 \(\overline{AC}?\)

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 \(56.\)

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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