A member posted this question and it lends itself to analytical methods quite well. Let's pretend you know nothing of those but you do know Geogebra!
2. Two poles, one 1m tall and one 2m tall are 3m apart. A length of wire is attached to the top of each pole and it is also staked to the ground somewhere between the two poles. Where should the wire be stacked so that the minimum amount of wire is used?
1) Put a point at (1,1) called A and a point at (4,2)called B.
2) Draw line segments from A to the x axis and B to the x axis. Label those points C and D. They represent the poles.
3) Place a point on the x axis between the poles called E. Place it at (2.5,0).
4) Draw line segments c and d from E to A and B ( the tops of the poles ).
5) Make sure in options point capturing is off.
6) Put into the Input bar c+d and press enter. e should appear in the variable pane and be about 4.29851. You might want to set rounding to 10.
7) Slide the point E on the x axis and watch e in the variable pane. Move it until you get the minimum value. See how close you get to 3√2 as the length of the wire and you estimate the position for the stake at (2,0). You should be able to eyeball this to 4 or 5 digits.
In mathematics, you don't understand things. You just get used to them.
I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.