Projection is not the same as realized

The pictures are hard to draw, yet the problem can be solved. Try to imagine how this looks like, or draw one of the several possible pictures. We will think of $\Pi_1$ and $\Pi_2$ as of horizontal and vertical planes respectively. Denote by $A_1,B_1,C_1,D_1$ the projections onto $\Pi_1$ and by $A_2,B_2,C_2,D_2$ the projections onto $\Pi_2.$ Since $|A_1B_1|=\sqrt{123},$ the difference of heights of $A$ and $B$ is $\sqrt{12^2-123}=\sqrt{21}.$ So the difference of heights between $A_1$ and $B_1$ is $\sqrt{21},$ thus the length of the projection of $A_1B_1$ to the line $l$ of intersection of the two planes is $\sqrt{123-21}=\sqrt{102}.$ Therefore, the difference of heights of $B_1$ and $C_1$ is $\sqrt{102},$ while the projection of $B_1C_1$ to the line $l$ has length $\sqrt{21}.$ By the Pythagorean Theorem $|BC|=\sqrt{102+123}=15.$ Continuing, $|CD|=12,$ $|DA|=15,$ so the perimeter of $ABCD$ is $54.$

Note: One can prove that $ABCD$ is a parallelogram.