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Square \(ABCD\) has area \(200\). Point \(E\) lies on side \(\overline{BC}\). Points \(F\) and \(G\) are the midpoints of \(\overline{AE}\) and \(\overline{DE}\), respectively. Given that quadrilateral \(BEGF\) has area \(34\), what is the area of triangle \(GCD\)?

 Feb 5, 2021

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We can use coordinates.  A = (0,sqrt(200)), B = (sqrt(200),sqrt(200)), C= (sqrt(200),0), D = (0,0).  Then G = (sqrt(200)/2,17), E = (sqrt(200),34), F = (sqrt(200),17), so the area of triangle GCD is 45.

 Feb 5, 2021

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