Calculating diagonal points over a distance

Question

Bit of an odd question but here goes...

I have got a fence 7m long. I have got some lights on string that is 10m long. I want to keep a uniformed look so need to put the lights in a zig zag pattern on the fence. How do i work out how many points i need across 7m long and the distance between points to keep it uniformed?

Otherwise i would have 3m of lights dangling down at the end of my fence

Does that make sense?!

Answer 1

If you were to create an isosceles single triangle with base 7m and total length 10m then you need

$$ h = \sqrt{ (5)^2 -(3.5)^2 } = 3.57 $$

Now with two triangles

$$ h = \sqrt{ (2.5)^2 - (1.75)^2 } = 2.38 $$

Or with $n$ triangles

$$ h = \sqrt{ \left( \frac{10}{n+1} \right)^2 - \left( \frac{7}{n+1} \right)^2 } $$

Finally to invert the above, given the fence height $h$

$$ n = \frac{ \sqrt{10^2-7^2}}{h}-1 =\frac{\sqrt{51}}{h}-1 $$