Length of hypotenuse v/s change in height of the opposite I have always struggled to understand mathematical concepts, and have a very different way of thinking about problems.  I suspect this is a very simple problem, but its confusing me a great deal.

I made a right triangle where the "opposite" side was $10$ and the "adjacent side" was $60$.  I then kept increasing the height of the opposite side by $10$ and measuring the length of the hypotenuse.  This rate of change in the hypotenuse length is not constant and reminds me of a sine wave because of its non-linear acceleration.

However, the rate of change appears to become linear as the graph progresses.

I just want to have an intuition for why the rate of change the length of the hypotenuse is not linear, and if there is one or several different ways I could visualize this to have a better intuition for whats going on.

Graph &amp; Triangle:
Imagine a right-angle triangle with opposite side of $x$, that you'll keep changing. An adjacent side $a$ a constant that you'll leave alone and fix. Then the length of the hypotenuse is given by $$\sqrt{x^2 + a}$$ 

Now, as $x$ grows very large, the constant $a$ becomes kind of irrelevant. Think of adding $10$ to $20,000,000,000,000$ it'll hardly make a difference to the calculuations, so we can say that $$\sqrt{x^2 + a} \approx \sqrt{x^2} = x$$
for large $x$. For the smaller values of $x$, that extra $+a$ will have quite an impact on the size of the $x^2 + a$ term.
This is a very good question and your diagram is helping me to understand a surveying problem im trying to solve, but anyhow i think i might be able to explain this a little bit, i hope haha.

Without getting into the math it might help to look at the problem by understanding the absolute limits of the triangle you are describing ( a right triangle with a fixed horizantal length) by finding the absolute limits of a geometric shape you will see that all real points that are obtainable fit within the absolute limits.  I hope this makes sense.  Absolute limits are 0 and infinty  both are un-attainable limits and everything must fit inside(or around in the case of zero) these limits.  Make sense?

So because of these two conditions you have created limits for the triangle.  When we take the two variables ( vertical length and hypotenuse length) to their limits ( infinity) we see that they both become infinite or in other words when the vertical length goes to infinity the hypotenuse goes to infinity at the same time.  This means that the vertical length is catching up to the hypotenuse length as the vertical gets longer ( hence not a linear ratio) until the vertical length reaches infinity and then it will equal the hypotenuse length of infinity.  Otherwise the vertical length is always less than the hypotenuse because infinty is un attainable.  This is why your graph stables out  vertical length is catching up to hypotenuse length the longer it gets.

Now, look at it another way.
If you take the vertical length to 0  you will see that the hypotenuse reaches a new limit of?.....  the adjacent side length.  And now when the vertical length is 0 it is no longer a triangle it is just a line.   As soon as the vertical length is anything greater than zero you will see that the hypotenuse becomes just greater than the adjacent length but it will never be less than or equal to the adjacent length or else you wont have a right triangle.   

Another limit of this triangle exist within the angle between the hypotenuse and the adjacent side.  This angle can go from anything greater than 0 to anything less than 90 degrees.  It will never be 0 or 90 degrees or else again the triangle breaks down and becomes a line. Even as that vertical line goes onto infinty the angle between the adjacent and hypotenuse will only approach 90 degrees but will never be 90 degrees. 

I hope this sheds some practical light on your mathmatical question. 

Cheers!