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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.

1 Answers 1

5

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 enter image description here

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    If you have the coordinates of the vertices, you have the lengths of the diagonals (and if you can use the determinant, it's even simpler). – 2017-01-27
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    @bernard the problem is i never learned the determinant or anything regarding matrices, so im unable to use that methos...... hence im using the the coordinates, which in this case r unknown since when i try to solve for them i get a undefined answer – 2017-01-27
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    Did you plot the $4$ lines? Don't forget the symmetry. – 2017-01-27
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    yes i did @bernard – 2017-01-27
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    @bernard the poi of the lines r not numbers on the graph ie.: poi of y=kx and y=(x-1)/k is............ (1/1-k^2, k/1-k^2) which is exxtreeemlllt hard to work with and then use pythagorean theorm with:(((( – 2017-01-27
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    You have a parallelogram in the third quadrant (if $k>0$), withh the origin as a vertex, an opposite vertex on the first bissectrix, and two vertices symmetric w.r.t. this bissectrix. Can't you solve for the systems of equations to obtain their coordinates? – 2017-01-27
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/52607/discussion-between-exchangehelpforuni-and-bernard). – 2017-01-27
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    @bernard hello again, may u plz post a pic or diagram as i stated in our chat – 2017-01-27
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    I'll can do that a little later – I have no way to scan here, so I'll have to make a latex picture for the figure. – 2017-01-27
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    what are the coordinates that u got? – 2017-01-27
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    Please see the plot I've added. The area is ½IJ×OL. – 2017-01-27
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    thanks for the image, however i do not understand what J O L are? Arent we dealing with ks and k+1? – 2017-01-27
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    @exchangehelpforuni: They are the intersection points of the lines of which you gave the equations. Sorry for not having seen your last comment before! – 2017-02-08