Let $k$ be a real number between $0$ and $1$.
What is the area of the quadrilateral formed by the lines $y = kx, y = kx + 1, x = ky$ and $x = ky + 1$?
I tried replacing $k$ with $0.5$, however it was hard to convert back. So what is an easy method that I can use to solve this problem? The method I used is so long. I tried graphing, however it is just a parallelogram.
Hint:
Plot the lines. Note the lines $y=kx$, $x=ky$ are symmetric w.r.t. the first bissectrix, and similarly for the lines $y=kx+1$, $x=ky+1$. Thus it is easy to have the coordinates of the vertices. Then remember the area of a parallelogram is (the absolute value of) a determinant.
Added: a plot of the figure