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?

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Uniteasy Admin · Accepted Answer

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 $$	riangle ODF$$, 
$$A_{OEQ}=A_{OGQ}$$ in $$	riangle OEG$$
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 $$35 cm^2 $$

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