Parallelogram inside a rectangle

Text

In the figure $ABCD$ is a rectangle with sides $8$ and $5$; the two oblique lines start from $D$ and from $B$ with a $45^\circ$ inclination. The dark figure is a parallelogram. Determine the area, perimeter and diagonal lengths of the parallelogram.

Solution

Set coordinates: $A(0;0)$, $B(8;0)$, $C(8;5)$, $D(0;5)$. The two $45^\circ$ lines are $D(0;5)\to(5;0)$ and $B(8;0)\to(3;5)$, both of slope $-1$: they are parallel. Vertical segments connect them. The vertices of the parallelogram are $P_1(3;2),\quad P_2(3;5),\quad P_3(5;3),\quad P_4(5;0).$ Sides. $\vec{P_1P_2}=(0;3)$ of length $3$; $\vec{P_2P_3}=(2;-2)$ of length $2\sqrt2$. Perimeter $2p=2\left(3+2\sqrt2\right)=6+4\sqrt2\approx11{,}66.$ Area (magnitude of the cross product of the sides from $P_1$, i.e. $(0;3)$ and $(2;-2)$): $\mathcal A=\bigl|0\cdot(-2)-3\cdot2\bigr|=6.$ Diagonals. $\vec{P_1P_3}=(2;1)\Rightarrow d_1=\sqrt5\approx2{,}24$; $\ \vec{P_2P_4}=(2;-5)\Rightarrow d_2=\sqrt{29}\approx5{,}39$. $\boxed{\mathcal A=6,\quad 2p=6+4\sqrt2\approx11{,}66,\quad d_1=\sqrt5,\quad d_2=\sqrt{29}.}$