Minimal Connector

Tommy from Homewood School and Sixth Form Centre in the UK found the total lengths of the first two networks and one shorter network:

Lisa and students from Hollingworth in South Africa worked on finding the minimum network that could connect the 4 points. They wrote:

We created a method much like those of the the students, but did it before looking at them. We labeled our small bar in the middle $x$, then created this equation:
$y = 4\sqrt{\left(\frac{10-x}2\right)^2+25}+x$

Then graphed it in Desmos and found the minimum x to be 4.226, with the minimum total length (y) as 27.321

Tommy used calculus to find the minimum network (click here to see a larger version):