Which of the following statements is false?

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(C) $1$) The area of a triangle depends on the coordinates of its vertices. Since translation of axes shifts the origin but keeps the axes parallel,th

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$3)$ The shifting of the origin to another point,without changing the direction of the axes,is called translation of axes.

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(C) $1$) The area of a triangle depends on the coordinates of its vertices. Since translation of axes shifts the origin but keeps the axes parallel,the relative distances and coordinates differences remain unchanged. Thus,the area is invariant. This statement is true.$2$) The slope of a line is defined by the ratio of the change in $y$ to the change in $x$. Under translation,$x = X + h$ and $y = Y + k$,so $dx = dX$ and $dy = dY$. The slope $m = \frac{dy}{dx} = \frac{dY}{dX}$ remains invariant. This statement is true.$3$) By definition,the translation of axes involves shifting the origin to a new point $(h, k)$ while keeping the axes parallel to the original axes. If the direction of the axes is changed,it is called rotation of axes. Therefore,this statement is false.$4$) If the origin is shifted to $(h, k)$,the new coordinates $(X, Y)$ are related to the old coordinates $(x, y)$ by $x = X + h$ and $y = Y + k$. Substituting these into the original equation $F(x, y) = 0$ gives $F(X+h, Y+k) = 0$. The statement provided is the inverse of this transformation logic. This statement is false.Note: In standard coordinate geometry,statement $3$ is false because translation does not involve changing the direction of the axes.

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