AMC10 2019 B

Answer (E): The reflection of the point $(a,b)$ across the line $y=x$ is the point $(b,a)$. Because the coordinates of the points in the original triangle are all positive, it follows that the coordinates of the images will also be all positive. Thus (A) is always true. It is a property of reflections that the line segment connecting a point not on the line of reflection and its image is perpendicular to the line of reflection. This fact shows that (C) and (D) are always true. Reflection is a rigid transformation, and therefore areas are preserved, so (B) is always true. The statement (E) is not true in general. As an example, consider a triangle $ABC$ such that the side $AB$ is parallel to the line $y=x$. Then the side $A'B'$ in the image will also be parallel to $y=x$, which shows that lines $AB$ and $A'B'$ are not perpendicular. Thus the statement (E) is not always true. In fact, because reflection across the line $y=x$ interchanges the roles of $x$ and $y$, the slope of a non-vertical/non-horizontal line and the slope of its image are reciprocals, not negative reciprocals. A line will be perpendicular to its reflection across the line $y=x$ if and only if the line is horizontal or vertical, in which case its image will be vertical or horizontal, respectively.