D, E, F, G are midpoints of sides of square ABCD. Given the areas of the three quadrilaterals, what is the area of the shaded one?

Connect DF, EG, interesting at point GQ. Apparently, Q is the midpoint of DF and EG.

Connect OQ. So we get,

A_{ODQ}=A_{OFQ} in \triangle ODF,

A_{OEQ}=A_{OGQ} in \triangle OEG

Let T be the area of the square ABCD

A_{OFCG}+A_{ODBE}

=(\dfrac{1}{4}T+A_{OGQ}+A_{OFQ}) + (\dfrac{1}{4}T-A_{ODQ}-A_{OFQ})

=\dfrac{1}{2}T

=27+45

=73

Similarly,

A_{ODAG}+A_{OECF} = \dfrac{1}{2}T

A_{OECF} = \dfrac{1}{2}T - A_{ODAG}

=73-38

=35

Therefore, the area of the shaded region is 35 cm^2