Given the diagram below, I'm trying to determine the value of $x$ in terms of $a$, $b$ and $c$.
I've already run this with a few of my colleagues. It seems obvious that the problem has only one solution, but we can't find it… is trigonometry the way to go, as we thought?
If anyone is interested, the original problem I was trying to solve was this: I was tracing a letter Z (in Inkscape) and wanted to make sure the stroke width was constant. In order to do this, I need to determine the horizontal and vertical positions for the start and end of the diagonal bar, as illustrated here:
Best Answer
The trick here is to shift the perpendicular between the two diagonal lines upwards so that two similar triangles are obtained. Then, by applying the Pythagorean theorem, we obtain $$\frac ax=\frac b{\sqrt{b^2+(c-x)^2}}$$ which may be manipulated to obtain a quadratic polynomial for $x$: $$a\sqrt{b^2+(c-x)^2}=bx$$ $$a^2(b^2+(c-x)^2)=b^2x^2$$ $$b^2+c^2-2cx+x^2=\frac{b^2}{a^2}x^2$$ $$\left(1-\frac{b^2}{a^2}\right)x^2-2cx+b^2+c^2=0$$ After solving, you need to check whether the obtained $x$ values are sensible – they have to lie within 0 and $c$.