In the figure shown below, square \(MNPT\) is inscribed in square \(ABCD\). The length of \(\overline{DC}\) is \(x\) inches, and the length of \(\overline{BN}\) is \(y\) inches. In terms of \(x\) and \(y\), which of the following expressions gives the area, in square inches, of \(MNPT\) ?

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The sides of the smaller square are the hypotenuse of the triangles. The triangles have sides \(x-y\) and \(y\). Using the pythaogrean theorem:

$$ a^2+b^2=c^2 $$ $$ (x-y)^2+y^2 = c ^2 $$ $$ =x^2-2xy+y^2+y^2 = c^2 $$ $$ c^2=\boxed{x^2-2xy+2y^2} $$

Since the area of a square is the side squared, \(c^2\) equals the area of the smaller square.