This is a great question. Love it love it love it!!
The following is just what occurs to me off the top of my head.
Show a graph paper grid with a dot at the origin and a dot at (3,4). Say we want to find the distance between the dots. We could guess that it would be 3+4=7. Well, that would be right if these were city blocks, but it's not the right answer as the crow flies. Roughly what do you think the distance is as the crow flies?
OK, let's measure it. Could someone draw it on the board at a scale where the coordinates are 30 cm and 40 cm? Thanks. What do you get when you measure it? 49 cm? Can you estimate the millimeters? I know it's rough. OK, you got 49.3 cm.
So now we know that the formula x+y isn't right. Actually, our answer looks really close to 50 cm. We don't know who the first person was to notice this, but supposedly it was already known to the ancient Egyptians. (Picture of rope with knots.) Yes, the conjecture is right, the true answer really is 5, but the Egyptians never knew why.
Here's one thing we can tell for sure about the mystery formula that gives the distance in terms of x and y. Suppose I draw a circle with a radius of 5, centered on the origin. It passes through our point from before. Now look at the top of the circle. It looks really flat there. If you were to zoom in on that point under an infinitely powerful microscope, it would look perfectly flat. You wouldn't even be able to see that it was curved, just like you can't tell that the earth is round when you're standing on the earth.
This is a hint that the relationship is nonlinear, meaning that it isn't just about ordinary addition and subtraction, like x+y or 2x+3y or something like that. I'm going to give you a preview of the actual answer, and then we'll come back and prove it. People probably did this originally as a guess and check (or guess and measure), and may never have been able to give an actual proof until hundreds of years later.
So the actual expression relating the distance d to x and y is $d^2=x^2+y^2$. This is called nonlinear because it has the exponents in it. When you square a big number, it gets really big. When you square a small number, it gets really small by comparison. What do you get when you square 1000? What do you get when you square 0.1?
This makes sense in terms of the close-up flatness idea. At a point like this (point to somewhere close to the top of the circle), x is really small. The square of a small number is really small, so $x^2$ is so small, we hardly care about it. So $d^2\approx y^2$. Can anyone solve this equation for y?
[...continue from here, lead in to formal proof...]