OofPirate

Thanks! It was correct! I had the concept of the conjugate from complex numbers, but I didn't know much about the variables itself!

Turns out that the Microsoft scientific calculator is a powerful tool. You can calculate 2^2012, then use mod 10 on it to get the answer of: 

nope not going to tell you Try it!

I like how you assigned coordinates to each of the vertices! This is very useful for coordinate proofs! I should use that in the future

The triangle with the side length 4 also has a height of 15-3, which is 12. Using the Pythagorean theorem, the hypotenuse is 4sqrt10. 

On the other hand, we can split the quadrilateral on the left into a rectangle and a triangle, and that triangle has a base of 14-4=10 and a height of 15-3=12. Using the Pythagorean theorem, we get that the hypotenuse of that triangle is 2sqrt61.

Therefore, the length of the rope is $2\sqrt{61}+4\sqrt{10}$

True

oh! and forgot to mention: For that tool link, there's a number conversion table, so you can compare all bases up to base 36. 

http://www.unitconversion.org/unit_converter/numbers-ex.html

I didn't use an algebraic way, but I used a base change calculator, so essentially guess and check.

However I knew that the base had to be above 11, since B is not included in the Base 11 digits. 

Here's a good tool: http://www.unitconversion.org/unit_converter/numbers-ex.html

sorry, that was incorrect. Is there a way to do this problem?

Edit: Nevertheless, I realized I made a calculation mistake, and I had found that the answer is a=12

Thanks! Have a nice day!