Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Let the rectangle ABCD initially lie in the xy-plane with B at the origin, BC along the x-axis and BA along the y-axis: $A(0,2r,0),C(r,0,0),D(r,2r,0)$. Equation of BD is $y=2x$; the foot N of C on BD is $\left(\frac{r}{5},\frac{2r}{5}\right)$ and $CN=\frac{2r}{\sqrt{5}}$. Clearly $CN=C_{1}N$, with $C_{1}N$ along the z-direction after folding, so $C_1\left(\frac{r}{5},\frac{2r}{5},\frac{2r}{\sqrt{5}}\right)$. Now $A{C}_1^2=\frac{r^2}{25}+\frac{64r^2}{25}+\frac{4r^2}{5}=\frac{85r^2}{25}\Rightarrow A{C}_1=\frac{\sqrt{85}r}{5}$.