2019 AMC 10B Problem 5

Reflection across $y=x$ sends $(x,y)$ to $(y,x)$. It therefore preserves the first quadrant and preserves area, so A and B are always true. The direction from $(x,y)$ to its image $(y,x)$ is $(y-x,x-y)$, whose slope is $-1$ because $x\ne y$. Thus C and D are always true as well. For E, take $A=(2,1)$, $B=(3,2)$, and $C=(4,1)$. These points satisfy all the conditions, but $AB$ and $A'B'$ both have slope $1$, so the two lines are parallel rather than perpendicular. Thus, the answer is E .